text:
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New Kaira D/A School: Pupil in the Volta Region sits Outside to Study Whiles new Classrooms Remain Locked
Tuesday, August 10, 2021 at 4:24 PM by Linda Anderson
Pupils of New Kaira D/A school are being forced to study on the corridors of their classroom blocks as the classrooms have been locked for almost a year after renovation
In an interview with some of the pupils, they lamented about the situation and how they suffer when it rains
The headmaster of the school stated that all efforts to get the classroom blocks opened have proved futile
Trending topics on the go: How we write news at YEN.com.gh
Pupils of the New Kaira School in the Volta Region are reported to be studying in the open whiles renovated school block remains locked for almost a year.
The school was established in the year 2000 but took close to 20 years to be given a fix, TV3 News reported.
The renovation since 2020 has been completed but the is the project is yet to be handed over to the school authority.
New Kaira D/A School: Pupil in the Volta Region sits Outside to Study Whiles new Classrooms Remain Locked Source: TV3 News
Students are now compelled to study from the corridors of the school.
Speaking to one of the pupils, it was revealed that they suffer a lot especially when it's raining.
Another student stated, they are not happy, they want to enter the classroom to learn.
Students are the nursery have no classrooms at all and would continue to study under trees even after the classroom blocks are opened.
A teacher by the name, Lydia Quenum added that the situation has affected student performance and some have had to drop out of school.
Headteacher of the school, Daniel Dzube intimated that all efforts to get the classrooms opened have not been successful.
YEN.com.gh earlier reported that a touching video has gathered massive reactions on social media as it showed the moment a group of students in Ghana studying under trees, celebrated their 'Our Day'
The footage shared by the famous Teacher Kwadwo, which ended up going viral, saw the students looking happy whilst they gulped down what appeared to be rice without meat or fish.
Although they could not afford much, the students were seen managing the little they had with great joy, which has warmed many hearts on social media.
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text:
\section{Introduction}

Substructural logic is logic lacking some of the well-known structural rules such as contraction, weakening, or exchange. Algebraically, substructural logics represent ordered residuated algebras \cite{galatos2007residuated}. One may examine modalities in substructural logics at least in two perspectives. Modalities in those logics are pretty helpful is such philosophical issues such as a relevant necessity. The second perspective is more applied and related to such issues as resource management. Here, linear logic proved its efficiency in computer science and linguistics.

The computer science applications are more related to resource-sensitive computation and
related applications in type theory \cite{abramsky1993computational} \cite{girard1987linear}.

The noncommutative version of linear logic, the Lambek calculus \cite{lambek1958mathematics}, has plenty of applications in linguistics, for instances, proof-theoretical characterisation of inference in Lambek grammars, the equivalent version of context-free grammars \cite{pentus1993lambek}. The Lambek calculus characterises inference in categorial grammars. There are several approaches to consider categorial grammars from a broader modal point of view as well, see the paper by van Benthem \cite{vanbenthem2003categorial}, for example.
Modalities in such logics (the $!$-modality, to be more precise) introduce lacking structural rules in a restricted way:

\begin{prooftree}
  \AxiomC{$\Gamma, \Delta \Rightarrow B$}
  \UnaryInfC{$\Gamma, ! A, \Delta \Rightarrow B$}
\end{prooftree}
\begin{minipage}{0.5\textwidth}
  \begin{flushleft}
    \begin{prooftree}
      \AxiomC{$\Gamma, ! A, B, \Delta \Rightarrow C$}
      \UnaryInfC{$\Gamma, B, ! A, \Delta \Rightarrow C$}
    \end{prooftree}
  \begin{prooftree}
  \AxiomC{$\Gamma, ! A, B, ! A, \Delta \Rightarrow C$}
  \UnaryInfC{$\Gamma, ! A, B, \Delta \Rightarrow C$}
  \end{prooftree}
  \end{flushleft}
\end{minipage}\hfill
\begin{minipage}{0.5\textwidth}
  \begin{flushright}
    \begin{prooftree}
      \AxiomC{$\Gamma, ! A, ! A, \Delta \Rightarrow B$}
      \UnaryInfC{$\Gamma, ! A, \Delta \Rightarrow B$}
    \end{prooftree}
  \begin{prooftree}
  \AxiomC{$\Gamma, ! A, B, ! A, \Delta \Rightarrow C$}
  \UnaryInfC{$\Gamma, B, ! A, \Delta \Rightarrow C$}
  \end{prooftree}
  \end{flushright}
\end{minipage}

  \vspace{\baselineskip}

Modal extensions of linear logic have an interpretation within the context of resource management based on the phase semantics proposed by Girard \cite{girard1995linear}. Algebraically, exponential modalities were studied by Ono as additional exponential operators on FL algebras \cite{ono1993semantics}. The Lambek calculus and its modal extensions also have the cover semantics proposed by Goldblatt \cite{goldblatt2011grishin} \cite{goldblatt2016cover}.
The abstract polymodal case of such an extension of the full Lambek calculus was recently studied by Kanovich, Kuznetsov, Scedrov, and Nigam \cite{kanovich2019subexponentials}. In this paper, (sub)exponential modalities are considered from proof-theoretical and complexity perspectives.

The basic Lambek calculus is complete with respect to so-called language models, residuated semigroups on subsets of free semigroup \cite{PENTUS1995179}. The Lambek calculus is complete with respect to subsets of a transitive relation as well, see \cite{Andreka1994}. Dunn, Gehrke, Palmigiano, and other authors considered relational semantics for the Lambek calculus using canonical extensions \cite{chernilovskaya2012generalized} \cite{dunn2005}. Alternatively, one may consider bi-approximation semantics for substructural logic studied by Suzuki \cite{Suzuki2010-SUZBSF-3}.

In this paper, we study the distributive version of the full Lambek calculus extended with normal modal operators $\Box$ and $\Diamond$ to consider a broader class of noncommutative modalities as an abstraction of storage operators in noncommutative linear logic in a distributive setting. We introduce noncommutative Kripke frames, relational structures for the distributive Lambek calculus extended with binary modal relations. We establish a discrete duality between such Kripke frames and perfect distributive residuated modal algebras developing an approach proposed in \cite{Distr}. After that, we overview canonical extensions of related modal algebras applying techniques provided in \cite{dunn2005} \cite{gehrketopological} \cite{Gehrke_Jonsson_2004} to show that any canonical residuated distributive modal logic is Kripke complete. We also prove that the subexponential modality axioms are canonical ones. Thus, we show that the corresponding logics enriched with subexponentials are Kripke complete. Finally, we extend the obtained duality to topological duality between residuated distributive modal algebras and special topological Kripke spaces based on bDRL-spaces, Priestley spaces with a ternary relation dual algebras of which are bounded distributive residuated lattices \cite{galatos2003varieties}. We also use some ideas from positive modal and intuitionistic modal logics \cite{celani1999priestley} \cite{palmigiano2004dualities}.

The text contains a short appendix with the brief survey on canonical extensions and duality for bounded distributive lattices to keep the paper self-contained.
\section{The distributive Lambek calculus with modal operators}
In this section, we formulate the distributive full Lambek calculus enriched with modal operators. We represent such logics with pairs of formulas that have the form $\varphi \vdash \psi$. One needs to read that as ``$\psi$ is derivable from $\varphi$''.

The language extends the language of the full Lambek calculus with modal operators $\Box$ and $\Diamond$ as follows:

$\varphi, \psi ::= \bot \:
   | \: \top \:
   | \: p \:
   | \: (\varphi \bullet \psi) \:
   | \: (\varphi \setminus \psi) \:
   | \: (\varphi / \psi) \:
   | \: (\varphi \lor \psi) \:
   | \: (\varphi \land \psi) \:
   | \: \Box \varphi \:
   | \: \Diamond \varphi$.

By a \emph{substructural distributive normal modal logic}, we mean some set of pairs $\vartheta \vdash \psi$, where $\vartheta$, $\psi$ are formulas, according to the following definition:
\begin{definition} A substructural normal distributive modal logic is a set of sequents $\Lambda$ that contains the axioms below and closed under the following inference rules:
\begin{center}
  $p \vdash p$
\end{center}
\begin{minipage}{0.5\textwidth}
  \begin{flushleft}
\begin{itemize}
  \item $\bot \vdash p$
  \item $p \vdash \top$
  \item $p_i \vdash p_1 \lor p_2, i = 1, 2$
  \item $p_1 \land p_2 \vdash p_i, i = 1, 2$
  \item $p \land (q \lor r) \vdash (p \land q) \lor (p \land r)$
  \item $(p \bullet q) \bullet r \dashv \vdash p \bullet (q \bullet r)$
\end{itemize}
\end{flushleft}
\end{minipage}\hfill
\begin{minipage}{0.5\textwidth}
  \begin{flushright}
\begin{itemize}
  \item $p \bullet {\bf 1} \dashv \vdash  {\bf 1} \bullet p \dashv \vdash p$
  \item $\Diamond (p \lor q) \vdash \Diamond p \lor \Diamond q$
  \item $\Diamond \bot \vdash \bot$
  \item $\Box p \land \Box q \vdash \Box (p \land q)$
  \item $\top \vdash \Box \top$
  \item $\Box p \bullet \Box q \vdash \Box (p \bullet q)$
\end{itemize}
\end{flushright}
\end{minipage}

\begin{minipage}{0.5\textwidth}
  \begin{flushleft}
    \begin{prooftree}
    \AxiomC{$\varphi \vdash \psi$}
    \AxiomC{$\psi \vdash \theta$}
    \BinaryInfC{$\varphi \vdash \theta$}
  \end{prooftree}

  \begin{prooftree}
    \AxiomC{$\varphi \vdash \psi$}
    \AxiomC{$\theta \vdash \psi$}
    \BinaryInfC{$\varphi \lor \theta \vdash \psi$}
  \end{prooftree}

  \begin{prooftree}
    \AxiomC{$\varphi \bullet \theta \vdash \psi$}
    \doubleLine
    \UnaryInfC{$\theta \vdash \varphi \setminus \psi$}
  \end{prooftree}

  \begin{prooftree}
    \AxiomC{$\varphi \vdash \psi$}
    \UnaryInfC{$\Box \varphi \vdash \Box \psi$}
  \end{prooftree}
  \end{flushleft}
\end{minipage}\hfill
\begin{minipage}{0.5\textwidth}
  \begin{flushright}
    \begin{prooftree}
    \AxiomC{$\varphi(p) \vdash \psi(p)$}
    \UnaryInfC{$\varphi [p := \gamma] \vdash \psi [p := \gamma]$}
  \end{prooftree}

  \begin{prooftree}
    \AxiomC{$\varphi \vdash \psi$}
    \AxiomC{$\varphi \vdash \theta$}
    \BinaryInfC{$\varphi \vdash \psi \land \theta$}
  \end{prooftree}

  \begin{prooftree}
    \AxiomC{$\theta \vdash \varphi \setminus \psi$}
    \doubleLine
    \UnaryInfC{$\varphi \bullet \theta \vdash \psi$}
  \end{prooftree}

  \begin{prooftree}
    \AxiomC{$\varphi \vdash \psi$}
    \UnaryInfC{$\Diamond \varphi \vdash \Diamond \psi$}
  \end{prooftree}
  \end{flushright}
\end{minipage}
\end{definition}

Substructural normal distributive modal logic extends distributive normal modal logic with residuals, product, and the axiom connecting $\Box$ and $\bullet$. See this paper \cite{Distr} to examine in-depth logics of bounded distributive lattices with modal operators that we extend.

We introduce ternary Kripke frames with the additional binary modal relations. Such a ternary frame might be considered as a noncommutative generalisation of a relevant Kripke, see, e.g., here \cite{Seki2003}. As it is usual in the relational semantics of substructural logic, product and residuals have the ternary semantics as in, e.g., \cite{Andreka1994} \cite{doi:10.1002/malq.19920380113} \cite{routley1973semantics}.

\begin{definition}
  A Kripke frame is a structure $\mathcal{F} = \langle W, \leq, R, R_{\Box}, R_{\Diamond}, \mathcal{O} \rangle$, where $\langle W, \leq, \rangle$ is a partial order, $R$ is a ternary relation on $W$, $R_{\Box}, R_{\Diamond}$ are binary relations on $W$, and $\mathcal{O} \subseteq W$ such that for all $\forall u, v, w, u', v', w' \in W$
  \begin{enumerate}
    \item $R u v w \: \& \: w R_{\Box} w' \Rightarrow \exists x, y \in W \:\:
    R x y w' \: \& \: u R_{\Box} x \: \& \: v R_{\Box} y$
    \item $\exists x \in W (R u w x \: \& \: R x u' v') \Leftrightarrow \exists y \in W (R w u' y \: \& \: R u y v')$
    \item $R u v w \: \& \: u' \leq u \Rightarrow R u' v w$, $R u v w \: \& \: v' \leq v \Rightarrow R u v' w$, $R u v w \: \& \: w \leq w' \Rightarrow R u v w'$
    \item $\forall o \in \mathcal{O} \:\: R v o w \Leftrightarrow R o v w $
    \item $v \leq w \Leftrightarrow \exists o \in \mathcal{O} R v o w$
    \item $\mathcal{O}$ is upwardly closed
    \item $u \leq v \: \& \: v R_{\Box} w \Rightarrow u R_{\Box} w$ and $u \leq v \: \& \: u R_{\Diamond} w \Rightarrow v R_{\Diamond} w$
  \end{enumerate}
\end{definition}

A Kripke model is a Kripke frame equipped with a valuation function that maps each propositional variable to
$\leq$-upwardly closed subset of worlds.

\begin{definition} Let $\mathcal{F} = \langle W, \leq, R, R_{\Box}, R_{\Diamond}, \mathcal{O} \rangle$ be a Kripke frame, a Kripke model is a pair $\mathcal{M} = \langle \mathcal{F}, \vartheta \rangle$, where $\vartheta : \operatorname{PV} \to \operatorname{Up}(W, \leq)$. Here, $\operatorname{Up}(W, \leq)$ is the collection of all upwardly closed sets. The connectives have the following semantics:

  \begin{enumerate}
    \item $\mathcal{M}, w \models p \Leftrightarrow w \in \vartheta(p)$
    \item $\mathcal{M}, w \models \top$, $\mathcal{M}, w \not\models \bot$, $\mathcal{M}, w \models {\bf 1} \Leftrightarrow w \in \mathcal{O}$
    \item $\mathcal{M}, w \models \varphi \bullet \psi \Leftrightarrow \exists u,v \in W \: R u v w \: \& \: \mathcal{M}, u \models \varphi \: \& \: \mathcal{M}, v \models \psi$
    \item $\mathcal{M}, w \models \varphi \setminus \psi \Leftrightarrow \forall u,v \in W \: R u w v \: \& \: \mathcal{M}, u \models \varphi \text{ implies } \mathcal{M}, v \models \psi$
    \item $\mathcal{M}, w \models \psi / \varphi \Leftrightarrow \forall u,v \in W \: R u w v \: \& \: \mathcal{M}, u \models \varphi \text{ implies } \mathcal{M}, v \models \psi$
    \item $\mathcal{M}, w \models \varphi \land \psi \Leftrightarrow \mathcal{M}, w \models \varphi \: \& \: \mathcal{M}, w \models \psi$
    \item $\mathcal{M}, w \models \varphi \lor \psi \Leftrightarrow \mathcal{M}, w \models \varphi \: \text{or} \: \mathcal{M}, w \models \psi$
    \item $\mathcal{M}, w \models \Box \varphi \Leftrightarrow \forall v \in R_{\Box}(w) \:\: \mathcal{M}, v \models \varphi$
    \item $\mathcal{M}, w \models \Diamond \varphi \Leftrightarrow \exists v \in R_{\Diamond}(w)
    \:\: \mathcal{M}, v \models \varphi$
    \item $\mathcal{M}, w \models \varphi \vdash \psi \Leftrightarrow \mathcal{M}, w \models \varphi \Rightarrow \mathcal{M}, w \models \psi$
    \item $\mathcal{M} \models \varphi \vdash \psi \Leftrightarrow \forall w \in W \: \mathcal{M}, w \models \varphi \vdash \psi$
  \end{enumerate}
\end{definition}

The following definitions are also standard ones.

\begin{definition}\label{frame}
  Let $\mathcal{F}$ be a Kripke frame
  \begin{enumerate}
    \item Let $\varphi \vdash \psi$ be a sequent, then $\mathcal{F} \models \varphi \vdash \psi$ iff for all valuations $\vartheta$ $\langle \mathcal{F}, \vartheta \rangle \models \varphi \vdash \psi$.
    \item $\operatorname{Log}(\mathcal{F}) = \{ \varphi \vdash \psi \: | \: \mathcal{F} \models \varphi \vdash \psi \}$.
    \item Let $\mathbb{F}$ be a class of Kripke frames, then $\operatorname{Log}(\mathbb{F}) = \bigcap \limits_{\mathcal{F} \in \mathbb{F}} \operatorname{Log}(\mathcal{F})$.
    \item Let $\Lambda$ be a substructural normal modal logic, then $\operatorname{Frames}(\mathcal{L}) = \{ \mathcal{F} \: | \: \mathcal{F} \models \mathcal{L} \}$ and $\mathcal{L}$ is \emph{complete} iff $\mathcal{L} = \operatorname{Log}(\operatorname{Frames}(\mathcal{L}))$.
  \end{enumerate}
\end{definition}
By ${\bf L}_{\bf K}$, we mean the minimal substructural distributive normal modal logic, the minimal set of sequents containing the axioms above and closed under the required inference rules.

The soundness theorem is the standard one.

\begin{theorem}
  Let $\mathbb{F}$ be a class of Kripke frames, then $\operatorname{Log}(\mathbb{F})$ is a substructural distributive normal modal logic.
\end{theorem}

\begin{proof} Let us show that $\mathcal{F} \models \Box p \bullet \Box q \Rightarrow \Box (p \bullet q)$, where $\mathcal{F} = \langle W, R, R_{\Box}, R_{\Diamond}, \mathcal{O}\rangle$ is an arbitrary frame. Let $\vartheta$ be a valuation and $\mathcal{M} = \langle \mathcal{F}, \vartheta \rangle$ a model such that $\mathcal{M}, w \models \Box p \bullet \Box q$, and $w \in W$.
Then there exists $u, v$ such that $R u v w$, $\mathcal{M}, u \models \Box p$, $\mathcal{M}, v \models \Box q$.
On the other hand, let $w' \in R_{\Box}(w)$. $R u v w \: \& \: w R_{\Box} w'$ implies $\exists x, y \in W \:\: R x y w' \: \& \: u R_{\Box} x \: \& \: v R_{\Box} y$. $\mathcal{M}, u \models \Box p$, $\mathcal{M}, v \models \Box q$ implies that for each $x \in R_{\Box}(u) \: \mathcal{M}, x \models p$ and for each $y \in R_{\Box}(v)$. $R x y w'$ implies $\mathcal{M}, w' \models p \bullet q$. Thus, $\mathcal{M}, w \models \Box(p \bullet q)$
\end{proof}

One may extend the notion of a bounded morphism for the relevant case to have homomorphisms between Kripke frames and models that preserve truth.

\begin{definition}
  Let $\mathcal{F}_1, \mathcal{F}_2$ be Kripke frames, a map $f : \mathcal{F}_1 \to \mathcal{F}_2$ is a bounded morphism, if:

  \begin{enumerate}
    \item $R_1 u v w \Rightarrow R_2 f(u) f(v) f(w)$
    \item $R_2 f(u) v' w' \Rightarrow \exists v, w \: v' \leq f(v) \: \& \: f(w) \leq w' \: \& \: R u v w$
    \item $R_2 u' f(v) w' \Rightarrow \exists u, w \: u' \leq f(u) \: \& \: f(w) \leq w' \: \& \: R u v w$
    \item $R_2 u' v' f(w) \Rightarrow \exists u, v \: u' \leq f(u) \: \& \: v' \leq f(v) \: \& \: R u v w$
    \item $f[(R_{\nabla})_1(x)] = (R_{\nabla})_2 (f(x))$, $\nabla = \Box, \Diamond$
    \item $f^{-1}[\mathcal{O}_2] = \mathcal{O}_1$
  \end{enumerate}
\end{definition}
By the notation $\mathcal{F}_1 \twoheadrightarrow \mathcal{F}_2$ we mean that there exists a surjective bounded morphism $f : \mathcal{F}_1 \twoheadrightarrow \mathcal{F}_2$ and we call such a map \emph{$p$-morphism}. Let $\mathcal{F}_1$, $\mathcal{F}_2$ be Kripke frames, $\vartheta_1$, $\vartheta_2$ valuations on $\mathcal{F}_1$ and $\mathcal{F}_2$ correspondingly, and $f : \mathcal{F}_1 \to \mathcal{F}_2$ a bounded morphism. Then $f : \mathcal{M}_1 \to \mathcal{M}_2$ is a bounded morphism of models, if
\begin{center}
  $\mathcal{M}_1, w \models p_i \Leftrightarrow \mathcal{M}_2, f(w) \models p_i$ for each propositional variable $p_i$.
\end{center}

The following lemma has the standard proof given in \cite{blackburn_rijke_venema_2001} \cite{chagrov1997modal}.

\begin{lemma}
$ $

  \begin{enumerate}
    \item $\mathcal{M}_1, w \models \varphi \Leftrightarrow \mathcal{M}_2, f(w) \models \varphi$.
    \item $\mathcal{F}_1 \twoheadrightarrow \mathcal{F}_2$ implies $\operatorname{Log}(\mathcal{F}_1) \subseteq \operatorname{Log}(\mathcal{F}_2)$.
    \item $\mathcal{F}_1 \cong \mathcal{F}_2$ implies $\operatorname{Log}(\mathcal{F}_1) = \operatorname{Log}(\mathcal{F}_2)$.
  \end{enumerate}
\end{lemma}

\section{Residuated distributive modal algerbas}

In this section, we study algebraic semantics and canonical extensions for substructural distributive modal logic. The required lattice-theoretic and canonical extensions definitions and notations are explained in the appendix. Let us define a residuated lattice \cite{Jipsen}.

\begin{definition}
  A residuated lattice is an algebra $\mathcal{R} = \langle \mathcal{L}, \cdot, \setminus, /, \varepsilon \rangle$, where $\mathcal{L}$ is a bounded lattice, $\cdot$ is a binary associative monotone operation,
  $\varepsilon$ is a multiplicative identity, $\setminus$ and $/$ are residuals, that is, the following equivalence holds for all $a, b, c \in \mathcal{L}$:

  \begin{center}
    $b \leq a \setminus c \Leftrightarrow a \cdot b \leq c \Leftrightarrow a \leq c / b$
  \end{center}
\end{definition}

Note that the class of all residuated lattices forms a variety since the quasi-identities above might be equivalently reformulated as identities, see \cite[Lemma 2.3]{Jipsen}.

A residuated lattice is called bounded distributive if its lattice reduct is a bounded distributive lattice. A residuated lattice morphism is a map $f : \mathcal{L}_1 \to \mathcal{L}_2$ that commutes with all operations in a usual way.

Let us recall the essential facts about (prime) filters on bounded distributive residuated lattices, see \cite{galatos2003varieties} \cite{Urq}. As a matter of fact, these statements hold for an arbitrary distributive lattice ordered semigroup since those properties of filters and their products do not depend on residuals.

\begin{lemma}\label{filter}
  Let $\mathcal{L}$ be a bounded distributive residuated lattice. Let $X, Y \subseteq \mathcal{L}$ and
  $X \cdot Y = \{ x \cdot y \: | \: x \in X, y \in Y \}$. Let us define $X \bullet Y = \uparrow X \cdot Y$, then
  \begin{enumerate}
    \item If $Z \subseteq \mathcal{L}$ is a filter, then $X \cdot Y \subseteq Z$ iff $X \bullet Y \subseteq Z$.
    \item If $X, Y \subseteq \mathcal{L}$ are filters, then $X \bullet Y$ is a filter.
    \item Let $X, Y, Z$ be filters in $\mathcal{L}$ and $Z$ is prime such that $XY \subseteq Z$, then there exist prime filters $X', Y'$ such that $X \subseteq X'$, $Y \subseteq Y'$, and $X' Y' \subseteq Z$.
  \end{enumerate}
\end{lemma}

A residuated distributive modal algebra is a bounded distributive residuated lattice extended with the operators $\Box$ and $\Diamond$ that distribute over finite infima and suprema respectively. One may also consider such algebras as full Lambek algebras \cite{ono1993semantics} \cite{Ono2019} reducts of which are bounded distributive lattices. Here, modalities are merely the ${\bf K}$-like operators without any additional requirements except the connection between $\Box$ and $\cdot$. We require that $\Box$ is also ``weakly normal'' with respect to the product. Such a ``weak normality'' corresponds to the promotion principle which is widespread in linear logic. This principle often has the ${\bf K}4$ form, where formulas in the premise are boxed from the left. This version of the promotion rule is rather the ${\bf K}$-rule than the ${\bf K}4$ one:

\begin{prooftree}
\AxiomC{$\varphi_1 \bullet \dots \bullet \varphi_n \vdash \psi$}
\UnaryInfC{$\Box \varphi_1 \bullet \dots \bullet \Box \varphi_n \vdash \Box \psi$}
\end{prooftree}

The inference rule also allows one to obtain the Kripke axioms formulated in terms of residuals as, e.g., $\Box(\varphi \setminus \psi) \Rightarrow \Box \varphi \setminus \Box \psi$ and $\Box (\psi / \varphi) \Rightarrow \Box \psi / \Box \varphi$.

This ``weak normality'' requirement is introduced as the additional inequation, more precisely:
\begin{definition}
  A residuated distributive modal algebra (RDMA) is an algebra $\mathcal{M} = \langle \mathcal{R}, \Box, \Diamond \rangle$ with the following conditions for each $a, b \in \mathcal{R}$:

  \begin{enumerate}
    \item $\Box (a \land b) = \Box a \land \Box b$, $\Box \top = \top$
    \item $\Diamond (a \lor b) = \Diamond a \lor \Diamond b$, $\Diamond \bot = \bot$
    \item $\Box a \cdot \Box b \leq \Box (a \cdot b)$
  \end{enumerate}

  An RDMA homomorphism is a bounded distributive residuated lattice homomorphism $f : \mathcal{M}_1 \to \mathcal{M}_2$ such that $f(\Box a) = \Box (f(a))$ and $f(\Diamond a) = \Diamond (f (a))$.
\end{definition}

One may associate with an arbitrary substructural normal modal logic its variety of RDMAs as follows:

\begin{definition}
  Let $\Lambda$ be a substructural normal modal logic, $\mathcal{V}_{\Lambda}$ is a variety defined by the set of inequations $\{ \varphi \leq \psi \: | \: \Lambda \vdash \varphi \vdash \psi \}$.
\end{definition}
Note that $\varphi$, $\psi$ are terms of the signature $\langle \wedge, \vee, \bot, \top, \cdot, \setminus, /, \varepsilon, \Box, \Diamond \rangle$ in such inequations as $\varphi \leq \psi$. One has an algebraic completeness for each substructural distributive normal modal logic as usual.

\begin{theorem}\label{linden}
  Let $\Lambda$ be a substructural normal modal logic, then there exists an RDMA $\mathcal{R}_{\Lambda}$ such that $\varphi \vdash \psi \in \Lambda$ iff $\mathcal{R}_{\Lambda} \models \varphi \leq \psi$.
\end{theorem}
One such RDMA is the free countably generated algebra in the variety $\mathcal{V}_{\Lambda}$, the Lindenbaum-Tarski algebra up to isomorphism.

The following statement also holds according to the general technique, see \cite{GOLDBLATT1989173}:

\begin{lemma}
  Let $\mathcal{\Lambda}$ be a residuated distributive normal modal logic, then the map $\Lambda \mapsto \mathcal{V}_{\Lambda}$ is the isomorphism between the lattice of residuated distributive normal modal logics and the lattice of varieties of RDMAs.
\end{lemma}

We define a completely distributive residuated perfect lattice as a distributive version of a residuated perfect one defined in \cite{dunn2005}.

\begin{definition}
  A distributive residuated lattice
  $\mathcal{L} = \langle L, \bigvee, \bigwedge, \cdot, \setminus, /, \varepsilon \rangle$ is called perfect distributive residuated lattice, if:

  \begin{itemize}
    \item Its lattice reduct is perfect distributive, see Definition~\ref{cdlattice}.
    \item $\cdot$, $\setminus$, and $/$ are binary operations on $L$ such that $/$ and $\setminus$ right and left residuals of $\cdot$, repsectively; $\cdot$ is a complete operator on $\mathcal{L}$, and $/ : \mathcal{L} \times \mathcal{L}^{\delta} \to \mathcal{L}$, $\setminus : \mathcal{L}^{\delta} \times \mathcal{L} \to \mathcal{L}$ are complete dual operators, where $\mathcal{L}^{\delta}$ is the dual of $\mathcal{L}$.
  \end{itemize}
\end{definition}

Here we formulate canonical extensions for bounded distributive lattices with a residuated family in the fashion of \cite{Gehrke2014}, where the author introduces canonical extensions for Heyting algebras. Here we take a generalised version of that construction formulated for residuated lattices, see \cite{gehrketopological}.

\begin{lemma}
  Let $\mathcal{L} = \langle L, \cdot, \setminus, /, \varepsilon \rangle$ be a bounded distributive residuated lattice, then $\mathcal{L}^{\sigma} = \langle L^{\sigma}, \cdot^{\sigma}, \setminus^{\pi}, /^{\pi}, \varepsilon \rangle$ is a perfect distributive residuated lattice.
\end{lemma}

Instead of proof that repeats this one \cite{gehrketopological}, we just define $\cdot^{\sigma}$, $\setminus^{\pi}$, and $/^{\pi}$ explicitly. Here we note that the canonical extension of a lattice reduct is a perfect distributive lattice \cite{gehrke1994bounded}.

Let $a, a' \in \mathcal{F}(\mathcal{L}^{\sigma})$ and $b \in \mathcal{F}(\mathcal{L}^{\sigma})$, then

\begin{enumerate}
  \item $a \setminus^{\pi} b = \bigvee \{ x \setminus y \: | \: a \leq x \in \mathcal{L} \ni y \leq b\}$ and similarly for the right residual
  \item $a \cdot^{\sigma} a' = \bigwedge \{ x \cdot x' \: | \: a \leq x \in \mathcal{L} \: \& \: a \leq x' \in \mathcal{L} \}$
\end{enumerate}

Let $a, b \in \mathcal{L}^{\sigma}$, then.
\begin{enumerate}
  \item $a \cdot^{\sigma} b =
  \bigvee \{ x \cdot^{\sigma} y \: | \: a \geq x \in \mathcal{F}(\mathcal{L}^{\sigma}) \: \& \: b \geq y \in \mathcal{F}(\mathcal{L}^{\sigma}) \}$
  \item $a \setminus^{\pi} b = \bigwedge \{ x \setminus^{\pi} y \: | \: a \geq x \in \mathcal{F}(\mathcal{L}^{\sigma}) \: \& \: b \leq y \in \mathcal{I}(\mathcal{L}^{\sigma})\}$ and $b /^{\pi} a$ is defined similarly
\end{enumerate}
The residuation property follows from the meet-density of $\mathcal{F}(\mathcal{L}^{\sigma})$ and join-density of $\mathcal{I}(\mathcal{L}^{\sigma})$ in $\mathcal{L}^{\sigma}$. Hence $\mathcal{L}^{\sigma}$ is a perfect distributive residuated lattice.

Let us describe the discrete duality for perfect distributive residuated lattices. Here we concretise the construction that establishes the discrete duality between perfect residuated lattices and perfect posets with ternary relation in \cite{dunn2005} within a distributive setting. We piggyback the Raney representation of perfect distributive lattices as algebras of downsets of completely join-irreducible elements \cite{raney1952completely} that generalise Birkhoff representation for finite lattices \cite{birkhoff1937rings}. We just recall that any perfect distributive lattice $\mathcal{L}$ is isomorphic to the lattice $\operatorname{Down}(\mathcal{J}^{\infty}(\mathcal{L}))$ mapping $a \in \mathcal{L}$ to $\mathcal{J}^{\infty}(\mathcal{L}) \cap \downarrow a$.

This representation might be extended to the duality between the categories of perfect distributive lattices and posets.

Let $\mathcal{L}$ be a perfect distributive residuated lattice. We define the relation $R \subseteq \mathcal{J}^{\infty}(\mathcal{L}) \times \mathcal{J}^{\infty}(\mathcal{L}) \times \mathcal{J}^{\infty}(\mathcal{L})$ as $R a b c \Leftrightarrow a \cdot b \leq c$.
Let us put $\mathcal{O} = \uparrow \varepsilon$, where $\varepsilon$ is a multiplicative identity. The structure $\mathcal{L}_{+} = \langle \mathcal{J}^{\infty}(\mathcal{L}), \leq, R, \mathcal{O} \rangle$ is the \emph{dual frame} of a perfect distributive residuated lattice $\mathcal{L}$.

Let $\langle W, \leq \rangle$ be a poset and $R \subseteq W^{3}$, $\mathcal{O}$ with the conditions (ii)-(vi) from Definition~\ref{frame}. Let us define the following operations on $\operatorname{Up}(W, \leq)$:
\begin{itemize}
  \item $A \setminus B = \{ w \in W \: | \: \forall u, v \in W \: R u w v \: \& \: u \in A \Rightarrow v \in B \}$
  \item $B / A = \{ w \in W \: | \: \forall u, v \in W \: R w u v \: v \in A \Rightarrow v \in B \}$
  \item $A \cdot B = \{ w \in W \: | \: \exists u, v \in W \: R u v w \: \& \: u \in A \: \& \: v \in B \}$
\end{itemize}

These operations are clearly well defined. Let us check $A \setminus B \in \operatorname{Up}(W, \leq)$, if $A, B \in \operatorname{Up}(W, \leq)$. Let $x \in A \setminus B$ and $x \leq y$, then $A \cdot \{ x \}\subseteq B$. Let $w \in \{ y \} \cdot A$, then there exists $a \in A$ such that $R y a w$, then $R x a w$ by the item (iii), Definition~\ref{frame}. Thus, $\{ y \} \cdot A \subseteq B$. Similarly, $B / A \in \operatorname{Up}(W, \leq)$. The residuation property for these operations holds immediately.

The following theorem establishes the discrete duality between perfect distributive residuated algebras and posets with a ternary relation that encodes the product. Let us call such a poset with the relation as above a \emph{ternary Kripke frame}.

\begin{theorem}\label{restheorem}
  $ $

  \begin{enumerate}
  \item Let $\mathcal{R}$ be a perfect distributive residuated lattice, then $\mathcal{R} \cong (\mathcal{R}_{+})^{+}$.
  \item Let $\mathcal{F}$ be a ternary Kripke frame, then $\mathcal{F} \cong (\mathcal{F}^{+})_{+}$.
\end{enumerate}
\end{theorem}

\begin{proof}
$ $

  \begin{enumerate}
    \item Let $R$ be a lattice reduct of $\mathcal{R}$. Accoding to the Raney representation, $\eta : R \cong \operatorname{Up}(\mathcal{J}^{\infty}(R), \leq_{\delta})$ such that $\eta : a \mapsto \{ b \in \mathcal{J}^{\infty}(R) \: | \: a \leq_{\delta} b\}$, where $\leq_{\delta}$ is a dual order on $\mathcal{J}^{\infty}(R)$. Let us ensure that this isomorphism also preserves products and residuals.

    Let $z \in \eta(a) \cdot \eta(b)$, then there exists $x \in \eta(a)$ and $y \in \eta(b)$ such that $x \cdot y \leq_{\delta} z$. That is, $a \leq_{\delta} x$ and $b \leq_{\delta} y$, so $a \cdot b \leq_{\delta} z$, then $z \in \eta(a \cdot b)$.

    Let $z \in \eta(a \cdot b)$, then $z$ is a join-irreducible element such that $a \cdot b \leq_{\delta} z$. Then $\uparrow z$ is a prime filter since $R$ is perfect distributive. It is clear that $\uparrow a \cdot \uparrow b \subseteq \uparrow b$. By Lemma~\ref{filter}, there exists prime filters $A$ and $B$ such that $\uparrow a \subseteq A$, $\uparrow b \subseteq B$,
    $A \cdot \uparrow b \subseteq \uparrow z$ and $\uparrow a \cdot B \subseteq \uparrow z$. $R$ is completely distributive, so there exists $a', b' \in \mathcal{J}^{\infty}(R)$ such that $\uparrow a' = A$ and $\uparrow b' = B$.
    Moreover, $\uparrow a \subseteq \uparrow a'$ and $\uparrow b \subseteq \uparrow b'$ implies $a \leq_{\delta} a'$ and $b \leq_{\delta} b'$, so $a' \in \eta(a')$ and $b' \in \eta(b)$. So $a' \cdot b' \leq_{\delta} z$, and, thus, $z \in \eta(a) \cdot \eta(b)$.

    $\eta$ preserves left and right residuals similarly to \cite[Lemma 6.10]{galatos2003varieties}. That was shown for arbitrary bounded residuated residuated lattices and the extended Priestley embedding.
    \item $\varepsilon : \langle W, \leq \rangle \to \langle \mathcal{J}^{\infty}(W, \leq), \leq_{\delta} \rangle$ is a poset isomorphism such that $\varepsilon : a \mapsto \uparrow a$. This isomorphism might be extended to the frame isomorphism via the frame conditions that connect a ternary relation with the partial order.
  \end{enumerate}
\end{proof}

\section{Discrete duality and completeness}

In this section, we establish a discrete duality between the categories of all Kripke frames and the category of all perfect residuated distributive modal algebras. We show that the Thomason theorem \cite{thomason1975categories} holds for normal residuated distributive modal logics.

\begin{definition}
  Let $\mathcal{L}$ be a perfect distributive residuated lattice and $\Box, \Diamond$ unary operators on $\mathcal{L}$, then
  $\mathcal{M} = \langle \mathcal{L}, \Box, \Diamond \rangle$ is called a perfect distributive residuated modal algebra, if for each where $A \subseteq \mathcal{L}$ and $a, b \in \mathcal{L}$
  \begin{itemize}
    \item $\Box \bigwedge A = \bigwedge \{ \Box a \: | \: a \in A \}$
    \item $\Diamond \bigvee A = \bigvee \{ \Diamond a \: | \: a \in A\}$
    \item $\Box a \cdot \Box b \leq \Box (a \cdot b)$
  \end{itemize}
\end{definition}
Given $\mathcal{M}$, $\mathcal{N}$ perfect residuated distributive modal algebras, a map $\mathcal{M} \to \mathcal{N}$ is a homomorphism if $f$ is a complete lattice homomorphism that preserves product, residuals, modal operators, and the multiplicative identity.

Let us show that the variety of all RDMAs is closed under canonical extensions.

\begin{lemma}\label{pRDMA}
  Let $\mathcal{R}$ be a istributive residuated lattice and $\mathcal{M} = \langle \mathcal{R}, \Box, \Diamond \rangle$ an RDMA, then $\mathcal{M}^{\sigma} = \langle \mathcal{R}^{\sigma}, \Box^{\pi}, \Diamond^{\pi} \rangle = \langle \mathcal{R}^{\sigma}, \Box^{\sigma}, \Diamond^{\sigma} \rangle$ is a perfect DRMA.
\end{lemma}

\begin{proof} The lattice reduct of $\mathcal{R}^{\sigma}$ is a perfect distributive lattice, \cite{gehrke1994bounded}.
In fact, one needs to show that the inequation $\Box a \cdot \Box b \leq \Box (a \cdot b)$ is canonical. Firstly, let us suppose that $a, b \in \mathcal{C}(\mathcal{M}^{\sigma})$. Note that $\Box^{\sigma} a \cdot^{\sigma} \Box^{\sigma} b = \bigwedge \{ \Box x \cdot \Box y \: | \: a \leq x \in \mathcal{M}, b \leq y \in \mathcal{M} \}$ that follows from the definition of a filter element, the fact that $\Box^{\sigma}$ preserves all infina and $\cdot^{\sigma}$ is an order-preserving operation. Then:

  $\begin{array}{lll}
  & \Box^{\sigma} a \cdot^{\sigma} \Box^{\sigma} b = & \\
  & \bigwedge \{ \Box x \cdot \Box y \: | \: a \leq x \in \mathcal{L}, \leq x \in \mathcal{L} \} \leq \bigwedge \{ \Box (x \cdot y) \: | \: a \leq x \in \mathcal{L} \: \& \: b \leq x \in \mathcal{L} \} = & \\
  & \bigwedge \Box^{\sigma} \{(x \cdot y) \: | \: a \leq x \in \mathcal{L} \: \& \: b \leq x \in \mathcal{L} \} = & \\
  & \Box^{\sigma} \bigwedge \{(x \cdot y) \: | \: a \leq x \in \mathcal{L} \: \& \: b \leq x \in \mathcal{L} \} = \Box^{\sigma} (a \cdot^{\sigma} b)&
  \end{array}$

  \vspace{\baselineskip}

  Let $a, b \in \mathcal{L}^{\sigma}$, then

  $\begin{array}{lll}
  & \Box^{\sigma} a \cdot^{\sigma} \Box b = \bigvee \{ \Box^{\sigma} x \cdot^{\sigma} \Box^{\sigma} y \: | \: a \geq x \in \mathcal{C}(\mathcal{L}^{\sigma}) \: \& \: b \geq y \in \mathcal{C}(\mathcal{L}^{\sigma}) \} \leq & \\
  & \bigvee \{ \Box^{\sigma} (x \cdot^{\sigma} y) \: | \: a \geq x \in \mathcal{C}(\mathcal{L}^{\sigma}) \: \& \: b \geq y \in \mathcal{C}(\mathcal{L}^{\sigma}) \} \leq & \\
  & \Box^{\sigma} \bigvee \{ x \cdot^{\sigma} y \: | \: a \geq x \in \mathcal{C}(\mathcal{L}^{\sigma}) \: \& \: b \geq y \in \mathcal{C}(\mathcal{L}^{\sigma}) \} = \Box^{\sigma} (a \cdot b)&
  \end{array}$
\end{proof}

\begin{definition}
  A substructural normal modal logic $\mathcal{L}$ is called canonical, if $\mathcal{V}_{\mathcal{L}}$ is closed under canonical extensions
\end{definition}

The complex algebra of a Kripke frame $\mathcal{F} = \langle W, \leq, R, R_{\Box}, R_{\Diamond}, \mathcal{O} \rangle $ is the complex algebra of the underlying residuated frame $\mathcal{F}^{+}$ with the modal operators defined as $[R_{\Box}] A = \{ u \in W \: | \: \forall w \: (u R_{\Box} w \Rightarrow w \in A)\}$ and $\langle R_{\Diamond} \rangle = \{ u \in W \: | \: \exists w \: (u R_{\Diamond w} \: \& \: w \in A)\}$. Here $A$ is upwardly closed subset. These operations are well-defined. The dual frame of a perfect RDMA $\mathcal{M} = \langle M, \bigvee, \bigwedge, \Box, \Diamond, \cdot, \setminus, /, \varepsilon \rangle$ is the dual frame $\mathcal{M}_{+}$ of an underlying perfect distributive residuated lattice with binary relations on completely join irreducible elements introduced as $a R_{\Box} b \Leftrightarrow \Box \kappa(a) \leq \kappa(b)$ and $a R_{\Diamond} b \Leftrightarrow a \leq \Diamond b$. Here, $\kappa$ is an order isomorphism between between $\mathcal{J}^{\infty}(\mathcal{M})$ and $\mathcal{M}^{\infty}(\mathcal{M})$.

Logically, Kripke frames and their complex algebras are connected with each other as follows:
\begin{proposition}\label{complex}
  Let $\mathcal{F}$ be a Kripke frame, then $\operatorname{Log}(\mathcal{F}) = \operatorname{Log}(\mathcal{F}^{+}) = \{ \varphi \vdash \psi \: | \: \mathcal{F}^{+} \models \varphi \leq \psi \}$
\end{proposition}

The following discrete duality theorem is merely a combination of Theorem~\ref{restheorem} and the similar fact proved for distributive modal algebras and frames for distributive modal logics \cite{Distr}.

\begin{theorem} \label{discrete}
$ $

  \begin{enumerate}
    \item Let $\mathcal{F}$ be a Kripke frame, then $\mathcal{F} \cong (\mathcal{F}^{+})_{+}$
    \item Let $\mathcal{M}$ be a perfect DRMA, then $\mathcal{M} \cong (\mathcal{M}_{+})^{+}$
    \item Functors $(.)_{+}: \operatorname{pDRMA} \rightleftarrows \operatorname{KF} : (.)^{+}$ establish a dual equivalence between the categories of all Kripke frames and all perfect RDMAs.
  \end{enumerate}
\end{theorem}

\begin{proof} It is easy to check that if $f : \mathcal{F}_1 \to \mathcal{F}_2$ is a bounded morphism of Kripke frames, then $f^{+} : {\mathcal{F}_2}^{+} \to {\mathcal{F}_1}^{+}$ such that $f^{+} : A \mapsto f^{-1}[A]$ is a perfect DRMA morphism. It is immediate that $h_{+} : {\mathcal{M}_2}_{+} \to {\mathcal{M}_1}_{+}$ is a bounded morphism, where $h : \mathcal{M}_1 \to \mathcal{M}_2$ is a perfect DRMA morphism. Thus, the dual equivalence follows from the previous two items and the lemma that claims that $(.)_{+}$ and $(.)^{+}$ are contravariant functors.
\end{proof}

The discrete duality established above together with canonical extensions of residuated distributive modal algebras provides the following consequence:

\begin{theorem}\label{complete}
  Let $\mathcal{L}$ be a canonical substructural distributive modal logic, then $\mathcal{L}$ is Kripke complete.
\end{theorem}

\begin{proof}
  The proof is similar to the analogous fact proved in \cite{Distr}, but we reproduce a sketch.

  Let $\varphi \vdash \psi \in \mathcal{L}$. Then $\mathcal{M}_{\mathcal{L}} \models \varphi \leq \psi$ by Theorem~\ref{linden}.
  But $\mathcal{M}_{\mathcal{L}} \models \varphi \leq \psi$ iff and only $(\mathcal{M}_{\mathcal{L}})^{\sigma} \models \varphi \leq \psi$ since $\mathcal{M}_{\mathcal{L}}$
  is a subalgebra of $(\mathcal{M}_{\mathcal{L}})^{\sigma}$ by Lemma~\ref{pRDMA} and the condition according to which $\mathcal{V}_{\mathcal{L}}$ is a canonical variety.
  By discrete duality, Theorem~\ref{discrete}, $({(\mathcal{M}_{\mathcal{L}})^{\sigma}}_{+})^{+} \models \varphi \leq \psi$.
  By Proposition~\ref{complex}, ${(\mathcal{M}_{\mathcal{L}})^{\sigma}}_{+} \models \varphi \vdash \psi$. In fact, ${(\mathcal{M}_{\mathcal{L}})^{\sigma}}_{+}$ is a canonical frame of a given logic $\mathcal{L}$ and we showed that ${(\mathcal{M}_{\mathcal{L}})^{\sigma}}_{+} \models \mathcal{L}$.
\end{proof}

As a consequence, the minimal normal substructural distributive modal logic is complete with respect to the class of all Kripke frames.
\begin{corollary}\label{completeK}
  ${\bf L}_{\bf K}$ is Kripke-complete.
\end{corollary}
Now we show that the following sequents that describe modalities as storage operators are canonical ones. This lemma partially repeats here \cite[Propositions 6.7 -- 6.10]{dunn2005}.
\begin{lemma}\label{canonseq} The following sequents are canonical:
  \begin{enumerate}
    \item $\Box p \bullet q \dashv \vdash q \bullet \Box p$
    \item $\Box p \vdash \Box p \bullet \Box p$
    \item $\Box p \bullet q \vdash \Box p \bullet q \bullet \Box p$, $q \bullet \Box p \vdash \Box p \bullet q \bullet \Box p$
    \item $\Box p \vdash \varepsilon$
    \item $\Box (p \land q) \dashv \vdash \Box p \bullet \Box q$
    \item $\Box (p \setminus q) \vdash \Diamond p \setminus \Diamond q$ and $\Box (p / q) \vdash \Diamond p / \Diamond q$
  \end{enumerate}
\end{lemma}

\begin{proof}
Let us check only the third sequent. The rest sequents might be checked similarly.

Let ${\bf K}_{nc}$ be ${\bf L}_{{\bf K}} \oplus \Box a \bullet b \vdash \Box a \bullet b \bullet \Box a$ and $\mathcal{V}_{{\bf K}_{nc}}$ its variety. Let us show that $\mathcal{V}_{{\bf K}_{nc}}$ is closed under canonical extensions.

Let $\mathcal{M} \in \mathcal{V}_{{\bf K}_{nc}}$, one needs to check that $\mathcal{M}^{\sigma} \in \mathcal{V}_{{\bf K}_{nc}}$. Let $a, b \in \mathcal{C}(\mathcal{M}^{\sigma})$.

  $\begin{array}{lll}
  & \Box^{\sigma} a \cdot^{\sigma} b = \bigwedge \{ \Box x \: | \: a \leq x \in \mathcal{M} \} \cdot^{\sigma} \bigwedge \{ y \: | \: b \leq y \in \mathcal{M} \} = & \\
  & \bigwedge \{ \Box x \cdot y \: | \: a \leq x \in \mathcal{M} \: \& \: b \leq y \in \mathcal{M} \} \leq & \\
  & \bigwedge \{ \Box x \cdot y \cdot \Box x \: | \: a \leq x \in \mathcal{M} \: \& \: b \leq y \in \mathcal{M} \} = \Box^{\sigma} a \cdot^{\sigma} b \cdot^{\sigma} \Box^{\sigma} a&
  \end{array}$

  \vspace{\baselineskip}

  Let $a, b \in \mathcal{M}^{\sigma}$, then

  $\begin{array}{lll}
  &\Box^{\sigma} a \cdot^{\sigma} b = \bigvee \{ \Box^{\sigma} x \cdot^{\sigma} y \: | \: a \geq x \in \mathcal{C}(\mathcal{M}^{\sigma}) \: \& \: b \geq y \in \mathcal{C}(\mathcal{M}^{\sigma}) \} \leq & \\
  & \bigvee \{ \Box^{\sigma} x \cdot^{\sigma} y \cdot^{\sigma} \Box^{\sigma} x \: | \: a \geq x \in \mathcal{C}(\mathcal{M}^{\sigma}) \: \& \: b \geq y \in \mathcal{C}(\mathcal{M}^{\sigma}) \} = \Box^{\sigma} a \cdot^{\sigma} b \cdot^{\sigma} \Box^{\sigma} a&
  \end{array}$
\end{proof}

The following completeness theorems follows from Theorem~\ref{complete}, Corollary~\ref{completeK}, and Lemma~\ref{canonseq}.

\begin{corollary}\label{structcompelete}
  Let $\Gamma$ be the set of all sequents from the lemma above and $\Delta \subseteq \Gamma$. Then $\mathcal{L} = {\bf L}_{{\bf K}} \oplus \Delta$ is Kripke complete.
\end{corollary}

One may also consider the subexponential polymodal case introduced in \cite{kanovich2019subexponentials}. Let us define a subexponential signature:

\begin{definition} A subexponential signature is an ordered quintuple: $\Sigma = \langle \mathcal{I}, \preceq, \mathcal{W}, \mathcal{C}, \mathcal{E} \rangle$,
where $\langle \mathcal{I}, \preceq \rangle$ is a preorder. $\mathcal{W}, \mathcal{C}, \mathcal{E}$ are upwardly closed subsets of $\mathcal{I}$ and $\mathcal{W} \cap \mathcal{C} \subseteq \mathcal{E}$.
\end{definition}

Let us define the following axioms:

\begin{itemize}
  \item $\Box_{s_1} p \bullet \Box_{s_2} q \vdash \Box_{s} (p \bullet q)$, $s \preceq s_1, s_2$
  \item $\Box_s p \vdash p$
  \item $\Box_s p \vdash \Box_s \Box_s p$
  \item $\Box_s p \bullet q \vdash \Box_s p \bullet q \bullet \Box_s p$, $q \bullet \Box_s p \vdash \Box_s p \bullet q \bullet \Box_s p$, where $s \in \mathcal{C}$
  \item $\Box_s p \bullet q \dashv \vdash p \bullet \Box_s q$, $s \in \mathcal{E}$
  \item $\Box_s p \vdash {\bf 1}$, where $s \in \mathcal{W}$
\end{itemize}

The system $\operatorname{DSMALC}_{\Sigma}$ is a substructural distributive polymodal logic with modal axioms as above plus $\Box_s (p \land q) \dashv \vdash \Box_s p \land \Box_s q$ and $\Box_s \top \dashv \vdash \top$ for each $s \in \mathcal{\Sigma}$. For simplicity, let us put the diamond axioms $\Diamond_s (p \lor q) \Leftrightarrow \Diamond_s p \lor \Diamond_s q$ and $\Diamond_s \bot \dashv \vdash \bot$ with the ${\bf S}4$ axioms for $\Diamond$ $p \vdash \Diamond_s p$ and $\Diamond_s \Diamond_s p \vdash \Diamond_s p$ for each $s \in \mathcal{\Sigma}$ axioms without any additional postulates. The modal inference rules have the form: from $\varphi \vdash \psi$ infer $\nabla_{s_2} \varphi \vdash \nabla_{s_1} \psi$ for $s_1 \preceq s_2$, where $nabla = \Box, \Diamond$.

\begin{theorem}
   $\operatorname{DSMALC}_{\Sigma}$ is canonical and, thus, Kripke complete.
\end{theorem}

\begin{proof}
  One may show that the variety $\mathcal{V}_{\operatorname{DSMALC}_{\Sigma}}$ is canonical showing that
  $\Box_{s_1} p \cdot \Box_{s_2} q \leq\Box_{s} (p \cdot q)$ for $s \preceq s_1, s_2$ is a canonical inequation similarly to Lemma~\ref{pRDMA}. The whole statement follows from the observation above and Theorem~\ref{complete}, Lemma~\ref{canonseq}, and Corollary~\ref{structcompelete}.
\end{proof}

\section{Topological duality}

In this section, we characterise a topological duality for residuated distributive modal algebras
in the same fashion as in
\cite{Esakia2019} \cite{Sambin1988TopologyAD}. We consider topological Kripke frames, ternary Kripke frames defined on Priestley spaces with binary modal relations, the category of which is dually equivalent to the category of all RDMAs. Alternatively, one may characterise such a duality in terms of general descriptive frames following the Goldblatt's approach \cite{goldblatt1993mathematics}. See the Appendix to have an explanation of the Priestley duality related definitions, terms, and notations.

Firstly, we consider a Priestley-style duality for residuated distributive bounded lattices. We piggyback the construction obtained by Galatos in his PhD thesis \cite{galatos2003varieties}. This construction is a noncommutative generalisation of relevant spaces, the dual spaces of relevant algebras studied by Urquhart \cite{Urq}. In fact, those spaces and their extensions with modal relations are the instances of relational Priestley spaces \cite{GOLDBLATT1989173}.

\begin{definition}
  Let $\mathcal{X} = \langle X, \tau, \leq \rangle$ be a Priestley space, $R \subseteq X^3$ and $E \subseteq X$.
  A bDRL-space is a tuple $\mathcal{X} = \langle X, \tau, \leq, R, E \rangle$ such that:

  \begin{enumerate}
    \item For all $x, y, z, w \in X$ there exists $u \in X$  such that $R(x,y,u)$ and $R(u,z,w)$ iff
    there exists $v \in X$ such that $R(y,z,v)$ and $R(x, v, w)$.
    \item For all $x, y, z, w \in X$ if $x \leq y$ then $R(y, u, v)$ implies $R(x, u, v)$,
    $R(u, y, v)$ implies $R(u, x, v)$, and $R(u, v, y)$ implies $R(u, v, x))$.
    \item Let $A, B \subseteq X$ be upwardly closed clopens then $R[A, B, \rule{0.25cm}{0.15mm} ]$, $\{ z \in X \: | \: R[z, B, \rule{0.25cm}{0.15mm}] \subseteq A\}$, and $\{ z \in X \: | \: R[B, z, \rule{0.25cm}{0.15mm} ]\}$ are also clopens.
    \item For all $x, y, z \in X$  $\neg R(x,y,z)$ iff there exists upwardly closed clopens $A, B \subseteq X$ such that that there exist $x \in A$ and $y \in B$ such that $z \notin R[A,B,\rule{0.25cm}{0.15mm}])$.
    \item $E$ is upwardly closed cloped such that if $A$ is clopen then $R[E, A, \rule{0.25cm}{0.15mm}] = R[A, E, \rule{0.25cm}{0.15mm}] = A$.
  \end{enumerate}

  Here $R[A, B, \rule{0.25cm}{0.15mm} ]$ denotes $\{ c \in X \: | \: \exists a \in A \: \exists b \in B \:\: R a b c \}$.
\end{definition}

We note that such a space is totally disconnected concerning a ternary relation according to the fourth condition. We introduce topological Kripke frames as bDRL-spaces with modal relations as follows.

\begin{definition} A \emph{modal bDRL-space} is a structure $\mathcal{X} = \langle X, \tau, \leq, R, R_{\Box}, R_{\Diamond}, E \rangle$, where $\langle X, \tau, \leq, R, E \rangle$ is a bDRL space and the following conditions hold:

\begin{enumerate}
\item If $A$ is upwardly closed clopen, then $R_{\Box}(A), R_{\Diamond}(A)$ are upwardly closed clopens.
\item For each $x \in X$, $R_{\Box}(x)$ and $R_{\Diamond}(x)$ are closed.
\item $R u v w \: \& \: w R_{\Box} w' \Rightarrow \exists x, y \in W \:\: R x y w' \: \& \: u R_{\Box} x \: \& \: v R_{\Box} y$.
\item $u \leq v \: \& \: v R_{\Box} w \Rightarrow u R_{\Box} w$.
\item $u \leq v \: \& \: u R_{\Diamond} w \Rightarrow v R_{\Diamond} w$.
\end{enumerate}
\end{definition}

Given a residuated distributive modal algebra $\mathcal{M} = \langle \mathcal{R}, \Diamond, \Box \rangle$ on a bounded distributive lattice $\mathcal{R}$, we define the set of all prime filters $\operatorname{PF}(\mathcal{L})$ and a map $\phi$ similarly to the bounded distributive lattice case described in the paper appendix. Let us define a ternary relation $R \subseteq \operatorname{PF}(\mathcal{L}) \times \operatorname{PF}(\mathcal{L}) \times \operatorname{PF}(\mathcal{L})$ as
$R(A, B, C) \Leftrightarrow A \bullet B \subseteq C$ and $E = \phi(\varepsilon)$. We also define binary relations $R_{\Box}$ and $R_{\Diamond}$ on $ \operatorname{PF}(\mathcal{L})$ as $A R_{\Diamond} B \Leftrightarrow b \in B \Rightarrow \Diamond b \in A$ and
$A R_{\Box} B \Leftrightarrow \Box a \in A \Rightarrow a \in B$.

Standardly, the subbasis of the topology $\tau$ is defined with by the sets $\phi(a)$ and $- \phi(a)$, $a \in \mathcal{M}$. Then the structure $\langle \operatorname{PF}(\mathcal{M}), \tau, E, R, R_{\Box}, R_{\Diamond} \rangle$ is the \emph{dual space} of a residuated distributive modal algebra $\mathcal{M}$.

Let $\mathcal{X} = \langle X, \tau, \leq, R, R_{\Box}, R_{\Diamond}, E \rangle$ be a modal bDRL-space and
$\operatorname{ClUp}(\mathcal{X})$ the set of all upwardly closed clopens of $\mathcal{X}$. We define product
as a binary operation on $\operatorname{ClUp}(\mathcal{X})$ as $A \circ B = R [A, B, \rule{0.25cm}{0.15mm}]$. Residuals are $A \setminus B = \{ c \in X \: | \: A \circ c \subseteq B \}$ and $B / A = \{ c \in X \: | \: c \circ A \subseteq B \}$. Modal operators $[R_{\Box}]$, $R_{\Diamond}$ are defined as $[R_{\Box}] A = \{ a \in X \: | \: \forall b \in X (a R_{\Box} b \Rightarrow b \in A)\}$
and $\langle R_{\Diamond} \rangle A = \{ a \in X \: | \: \exists b \in X (a R_{\Diamond} b \: \& \: b \in A)\}$.
The structure $\langle \operatorname{ClUp}(\mathcal{X}), \cup, \cap, \emptyset, X, \circ, \setminus, /, E, [R_{\Box}], \langle R_{\Diamond} \rangle \rangle$ is the \emph{dual algebra} of a modal bDRL-space $\mathcal{X}$. One may show that $\phi$ commutes with products and residuals \cite{galatos2003varieties}, that is:

$\phi$ commutes with modal operators as $\phi(\Diamond a) = \langle R_{\Diamond} \rangle [\phi(a)]$ and $\phi(\Box a) = [R_{\Box}] [\phi(a)]$ that also may be shown similarly to Proposition 5.2.1 here \cite{palmigiano2004dualities}.

The key theorem is the following one:

\begin{theorem}
$ $

\begin{enumerate}
\item The dual algebra of a modal bDRL-space $\mathcal{X}$ is an RDMA.
\item The dual space of a residuated distributive modal algebra $\mathcal{M}$ is a modal bDRL-space.
\end{enumerate}
\end{theorem}

\begin{proof}

\begin{enumerate}
  \item Let $A, B$ be upwardly closed clopens, let us show that $[R_{\Box}] A \bullet [R_{\Box}] B \subseteq [R_{\Box}] (A \bullet B)$.
  Let $c \in [R_{\Box}] A \bullet [R_{\Box}] B$, then $c \in R[R_{\Box}A, [R_{\Box}] B, \rule{0.25cm}{0.15mm}]$.
  This denotes that $R (a,b,c)$ for some $a \in R_{\Box} A$ and $b \in R_{\Box} B$. Let $c' \in X$ such that $c R_{\Box} c'$, let us show that $c' \in A \bullet B$.
  $R(a, b, c)$ and $c R_{\Box} c'$ implies that $R(a', b', c')$ for some $a' \in R_{\Box} (a)$ and $b' \in R_{\Box} (b)$ by the definition of a modal bDRL-space. Thus, $a' \in A$ and $b' \in B$, so $c' \in A \bullet B$.
  \item We check the condition that connect the ternary relation with the $\Box$-relation. Let $A, B$ be prime filters. Let us suppose that $R A B C$ and $C R_{\Box} C'$, where $A, B, C, C'$ are prime filters. Let us show that $R A' B' C'$ for some prime filters $A', B'$. Let us put $A_1 := \{ a \in X \: | \: \Box a \in A \}$ and $B_1 := \{ b \in X \: | \: \Box b \in B \}$. $A_1$ and $B_1$ are clearly filters. Now let us show that $A_1 \cdot B_1 \subseteq C'$. Let $a \cdot b \in A_1 \cdot B_1$, then $\Box a \in A$ and $\Box b \in B$. So $\Box a \cdot \Box b \in A \bullet B$, so $\Box (a \cdot b) \in A \bullet B$. Hence $\Box (a \cdot b) \in C$ and $a \cdot b \in C'$. By Lemma~\ref{filter}, there exist prime filters $A' \supseteq A_1$ and $B' \supseteq B_1$ such that $A'' B'' \subseteq C'$.

Let $A$ be a prime filter, then $R_{\Box} (A)$ and $R_{\Diamond}(A)$ are closed that might be shown as the well-known Esakia's lemma \cite{esakia1974topological}. Let $A$ be an upwardly closed clopen, $R_{\Box}(A)$ and $R_{\Diamond} (A)$ are upwardly closed clopens. The last two facts might be proved similarly to the intutionistic modal logic case \cite{palmigiano2004dualities}.
\end{enumerate}
\end{proof}

\begin{definition}
Given modal bDRL spaces $\mathcal{X}$, $\mathcal{Y}$, a contnuous bounded morphism is a map
$f : \mathcal{X} \to \mathcal{Y}$ such that $f$ is a Priestley map that preserves ternary and binary relations as a bounded morphism.
\end{definition}

As usual, the previous lemma allows one to claim that $\operatorname{PF}$ and $\operatorname{ClUp}$ are contravariant functors. The following theorem establish a desired topological duality itself.

\begin{theorem}
$ $

\begin{enumerate}
\item Let $\mathcal{M}$ be a residuated distributive modal algebra, then
$\operatorname{ClUp}(\operatorname{PF}(\mathcal{M})) \cong \mathcal{M}$
\item Let $\mathcal{X}$ be a modal dBRL-space, then
$\operatorname{PF}(\operatorname{ClUp}(\mathcal{X})) \cong \mathcal{X}$
\item Contravariant functors $\operatorname{PF} : \operatorname{RDMA} \rightleftarrows \operatorname{TKF} : \operatorname{ClUp}$ constitute a dual equivalence between the category of all substructural distributive modal algebras and the category of all modal dBRL-spaces.
\end{enumerate}
\end{theorem}

\begin{proof}
$ $

(i) The isomorphism is map $a \mapsto \phi(a)$ that commutes with products, residuals and modal operators as discussed above.

(ii) A homeomorphism is a map $g : x \mapsto \{ A \in \operatorname{ClUp} \: | \: x \in A \}$. As it is shown by Galatos,
$R_{\mathcal{X}}(x,y,z) \Leftrightarrow R_{\operatorname{PF}(\operatorname{ClUp}(\mathcal{X}))}(g(x), g(y), g(x))$.
One may immediately extend this homeomorphism and show that this map commutes with binary modal relations.

(iii) Follows from the previous two items and the previous theorem. Let us ensure briefly that these functors behave as expected with morphisms. Let $h : \mathcal{M}_1 \to \mathcal{M}_2$ be an RDMA homomorphism, then a map
$\operatorname{PF}(h) : \operatorname{PF}(\mathcal{M}_2) \to \operatorname{PF}(\mathcal{M}_1)$ such that
$\operatorname{PF}(h) : F \mapsto h^{-1} [F]$ is a Priestley map, where $F$ is a prime filter. This map also satisfies the monotonicity and
lifting properties for a ternary relation, see \cite{galatos2003varieties}. One may show that $\operatorname{PF}(h)$ has the monotonicity and lifting properties for $[R_{\Box}]$ and $\langle R_{\Diamond} \rangle$ similarly to the intutionistic modal logic case. On the other hand, $\operatorname{ClUp}(f) : \operatorname{ClUp}(Y) \to \operatorname{ClUp}(X)$ such that
$\operatorname{ClUp}(f) : A \mapsto f^{-1}[A], A \in \operatorname{ClUp}(Y)$ is a lattice homomorphism. $\operatorname{ClUp}(f)$
also preserves product and residuals. $\operatorname{ClUp}(f)$ preserves $[R_{\Box}]$ and $\langle R_{\Diamond} \rangle$
similarly to the intuitionistic modal logic case.
\end{proof}

Hence, we obtained the Priestley-style duality for the category of residuated distributive modal algebras and the category of all topological Kripke frames introduced by us.

\section{Further work}

In this paper, we examined canonicity for the distributive full Lambek calculus and its modal extensions within a ``usual'' Kripkean semantics. The further questions that should be solved are Sahlqvist and Goldblatt-Thomason theorems for such semantics and its non-distributive generalisation to study canonicity and modal definability for noncommutative modal logic with residuals in depth. One may consider for these purposes the frameworks described in \cite{conradie2018goldblatt} and \cite{DBLP:journals/apal/ConradieP19}.

One may also consider a Kleene star as a modal operator \cite{kuznetsov2018continuity}, but such a modality is neither $\Box$ nor $\Diamond$. The (non)canonicity of the variety of residuated Kleene lattices also should be studied and explored considering a Kleene residuated lattice as a sort of bounded lattice with operators.

We took the Lambek calculus with additive connections as the underlying logic requiring the lacking distributivity principle which is unprovable in the full Lambek calculus. A Priestley-style topological duality provided in this paper might be extended considering dual spaces for non-distributive residuated modal algebras using the canonical extensions technique studied by Gehrke and van Gool, e.g., here \cite{gehrke2014distributive}.

The distributive Lambek calculus also has a sort of Gentzen cut-free calculus and FMP \cite{kozak2009distributive}, but the same issues for the modal extensions are not investigated yet.

\Appendix

In this section, we recall the required background related to bounded distributive lattice canonical extensions and topological duality. Such notions as lattice, distributive lattice, filter, and prime filters are supposed to be known. We refer the reader to these textbooks
\cite{davey2002introduction} \cite{Sank}.

Let us recall Priestley duality, the dual equivalence between the caterory of bounded distributive lattices and the category of Priestley spaces \cite{priestley1972ordered}. First of all, we define a Priestley space.

\begin{definition}
  $ $

  \begin{enumerate}
  \item A Priestley space is a triple $\mathcal{X} = \langle X, \tau, \leq \rangle$, where $\langle X, \tau \rangle$ is a compact topological space, $\langle X, \leq \rangle$ is a bounded partial order with the additional Priestley separation axiom:
  \begin{center}
    $x \not\leq y \Rightarrow$ there exists a clopen $U \subseteq X$ such that $x \in U$ and $y \not\in U$
  \end{center}
  \item Let $\mathcal{X}_1, \mathcal{X}_2$ be Priestley spaces and $f : \mathcal{X}_1 \to \mathcal{X}_2$, then $f$ is a Priestley map, if $f$ is continuous, order-preserving and preserves bounds.
 \end{enumerate}
\end{definition}

Let $\mathcal{L}$ be a bounded distributive lattice and $\operatorname{PF}$ the set of prime filters in $\mathcal{L}$. Let us define a map $\phi : \mathcal{L} \to \mathcal{P}(\operatorname{PF}(\mathcal{L}))$ such that $\phi : a \mapsto \{ F \in \operatorname{PF}(\mathcal{L}) \: | \: a \in F \}$. The sets $\phi(a)$ and $- \phi(a)$ form a subbasis of topology on $\operatorname{PF}(\mathcal{L})$, where $a \in \mathcal{L}$. The structure $\langle \operatorname{PF}(\mathcal{L}), \tau, \subseteq \rangle$ is a Priestley space, where $\tau$ is generated by the subbasis above.

Let $\mathcal{X} = \langle X, \tau, \leq \rangle$ be a Priestley space and $\operatorname{ClUp}(\mathcal{X})$ the set of all clopen upwardly closed subsets of $\mathcal{X}$. The dual algebra of a Priestley space is an algebra $\langle \operatorname{ClUp}(\mathcal{X}), \cap, \cup, \emptyset, X \rangle$, which is a bounded distribute lattice.

Let $\mathcal{L}$ be a bounded distributive lattice, then
$\eta_{\mathcal{L}} : \mathcal{L} \to \operatorname{ClUp}(\operatorname{PF}(\mathcal{L}))$ is a lattice isomorphism. Given a Priestley space $\mathcal{X}$, then $\varepsilon_{\mathcal{X}} : \mathcal{X} \to \operatorname{PF}(\operatorname{ClUp}(\mathcal{X}))$ is a Priestley homeomorphism. Moreover, if $h : \mathcal{L}_1 \to \mathcal{L}_2$ is a bounded lattice homomorphism,
then $\operatorname{PF}(h) : \operatorname{PF}(\mathcal{L}_2) \to \operatorname{PF}(\mathcal{L}_1)$ is a Priestley map, where $\operatorname{PF}(h) : A \mapsto h^{-1}(A)$. If $f : \mathcal{X}_1 \to \mathcal{X}_2$ is a Priestley map, then $\operatorname{ClUp}(f) : \operatorname{ClUp}(\mathcal{X}_2) \to \operatorname{CU}(\mathcal{X}_1)$ is a bounded lattice homomorphism, where $\operatorname{ClUp}(f) : A \mapsto f^{-1}(A)$. The facts mentioned above establish Priestley duality:

\begin{theorem}
  The functors $\operatorname{PF} : {\bf BDistr} \rightleftarrows {\bf Priest} : \operatorname{ClUp}$ constitute a dual equivalence between the category of all bounded distributive lattices and the category of all Priestley spaces.
\end{theorem}

  \vspace{\baselineskip}

Canonical extensions were introduced by Jonsson and Tarski to extend a Stone representation to Boolean algebras with operators \cite{BAOs}.
Let us overview canonical extensions of distributive lattice expansions. We refer the reader to these paper \cite{gehrke1994bounded} \cite{Gehrke_Jonsson_2004} to have a more detailed picture of bounded distributive lattice expansions and canonical extensions for them.

Given a complete lattice $\mathcal{L}$, $a \in \mathcal{L}$ is called \emph{completely join irreducible},
if $a = \bigvee \limits_{i \in I} a_i$ implies that $a = a_i$ for some $i \in I$. Completely meet irreducible elements are defined dually. By $J^{\infty}(\mathcal{L})$ ($M^{\infty}(\mathcal{L})$ we denote the set of all completely join (meet) irreducible elements. There is an order isomorphism $\kappa : J^{\infty}(\mathcal{L}) \to M^{\infty}(\mathcal{L})$ such that $\kappa : u \mapsto \bigvee (- \uparrow u)$ \cite{gehrke1994bounded}.

A complete lattice $\mathcal{L}$ is \emph{completely distributive} \cite{davey2002introduction}, if for each doubly indexed subset $\{ x_{ij} \}_{i \in I, j \in J}$ of $\mathcal{L}$ one has
\begin{center}
  $\bigwedge_{i \in I} (\bigvee_{j \in J} x_{ij}) = \bigvee_{\alpha : I \to J} (\bigwedge_{i \in I} x_{i\alpha(i)})$
\end{center}
A perfect distributive lattice is a doubly algebraic completely distributive lattice, that is:
\begin{definition} \label{cdlattice}
  Let $\mathcal{L}$ be a bounded distributive lattice, then $\mathcal{L}$ is called perfect distributive lattice, if it is completely distributive and every $x \in \mathcal{L}$ has the form $x = \bigvee \{ j \in J^{\infty}(\mathcal{L}) \: | \: j \leq x\}$ and
  $x = \bigwedge \{ m \in M^{\infty}(\mathcal{L}) \: | \: x \leq m \}$. That is,
  $J^{\infty}(\mathcal{L})$ and $M^{\infty}(\mathcal{L})$ are join-dense
  and meet-dense in $\mathcal{L}$ correspondingly.
\end{definition}

Given a lattice $\mathcal{L}$, by the completion of $\mathcal{L}$ we mean an embedding
$\mathcal{L} \hookrightarrow \mathcal{L}'$, where $\mathcal{L}'$ is a complete lattice. For simplicity, we assume that $\mathcal{L}'$ contains $\mathcal{L}$ as a sublattice.
The definition of a canonical extension is the standard one:
\begin{definition}
  Let $\mathcal{L}$ be a bounded distributive lattice, a canonical extension of $\mathcal{L}$ is a completion
  $\mathcal{L} \hookrightarrow \mathcal{L}^{\sigma}$, where $\mathcal{L}^{\sigma}$ is
  a complete lattice and the following conditions hold:
  \begin{enumerate}
    \item (Density) Every element of $\mathcal{L}^{\sigma}$ is both a join of meets and meets of joins of elements from $\mathcal{L}$
    \item (Compactness) Let $S, T \subseteq \mathcal{L}$ such that $\bigwedge S \leq \bigvee T$ in $\mathcal{L}^{\sigma}$, then there exist finite subsets $S' \subseteq S$ and $T' \subseteq T$ such that $\bigwedge S' \leq \bigvee T'$.
  \end{enumerate}
\end{definition}
Now we define filter and ideal elements, or, closed and open elements, according to the alternative terminology.
\begin{definition}
    Let $\mathcal{L}$ be a bounded distributive lattice and $\mathcal{L}^{\sigma}$ a canonical extension of $\mathcal{L}$. Let us define the following sets:
    \begin{enumerate}
      \item $\mathcal{F}(\mathcal{L}^{\sigma}) = \{ x \in \mathcal{L}^{\sigma} \: | \: \text{$x$ is a meet of elements from $\mathcal{L}$}\}$, the set of filter elements
      \item $\mathcal{I}(\mathcal{L}^{\sigma}) = \{ x \in \mathcal{L}^{\sigma} \: | \: \text{$x$ is a join of elements from $\mathcal{L}$}\}$, the set of ideal elements
    \end{enumerate}
\end{definition}
It is known that the poset $F(\mathcal{L}^{\delta})$ is isomorphic to the poset $\operatorname{Filt}(\mathcal{L})$, the set of all filters of $\mathcal{L}$ and the similar statement holds for $\mathcal{I}(\mathcal{L}^{\sigma})$ and the of all ideals of $\mathcal{L}$.
We recall that a canonical extension of a bounded lattice $\mathcal{L}$ is unique up to isomoprphism that fixes $\mathcal{L}$, see, e. g., \cite{Expan}. For each canonical extension $\mathcal{L} \hookrightarrow \mathcal{L}^{\sigma}$, the poset $\mathcal{F}(\mathcal{L}^{\sigma}) \cup \mathcal{I}(\mathcal{L}^{\sigma})$ is uniquely defined by $\mathcal{L}$. The uniqueness of a canonical extension of $\mathcal{L}$ up to an isomorphism fixing $\mathcal{L}$ follows from this observation.

In the case of bounded distributive lattices, one has the following fact \cite{gehrke1994bounded}:
\begin{prop}
  $ $

  Let $\mathcal{L}$ be a bounded distributive lattice, then $\mathcal{L}^{\sigma}$ is a perfect distributive lattice.
\end{prop}

We also note that canonical extensions commute with dual order and Cartesian product. That is, $(\mathcal{L}^{op})^{\sigma} \cong (\mathcal{L}^{\sigma})^{op}$ and $(\mathcal{L}^n)^{\sigma} \cong (\mathcal{L}^{\sigma})^n$. Now we overview bounded distributive lattices expansions, that is,
bounded distributive lattices enriched with the additional family of operators, and their canonical extensions \cite{Expan}.

  Let $\mathcal{K}$, $\mathcal{L}$ be bounded distributive lattices and
$f : \mathcal{K} \to \mathcal{L}$.
Let us define maps $f^{\sigma}$, $f^{\pi} : \mathcal{K}^{\sigma} \to \mathcal{L}^{\sigma}$
as follows:
\begin{enumerate}
    \item $f^{\pi}(u) = \bigwedge \{ \bigvee \{ f(a) \: | \: a \in \mathcal{K}, x \leq a \leq y \} \: | \: \mathcal{F}(\mathcal{K}^{\sigma}) \ni x \leq u \leq y \in \mathcal{I}(\mathcal{K}^{\sigma}) \}$
    \item $f^{\sigma}(u) = \bigvee \{ \bigwedge \{ f(a) \: | \: a \in \mathcal{K}, x \leq a \leq y \} \: | \: \mathcal{F}(\mathcal{K}^{\sigma}) \ni x \leq u \leq y \in \mathcal{I}(\mathcal{K}^{\sigma}) \}$
\end{enumerate}
Every element of $\mathcal{K}$ is a filter element and an ideal element of $\mathcal{L}^{\sigma}$,
then $f^{\sigma}$ and $f^{\pi}$ both extend $f$. It is clear that $f^{\sigma} \leq f^{\pi}$ in a pointwise order. We formulate the following fact about extensions of maps on bounded distributive lattices that we are going to use further. If the original map $f$ is order preserving, then one may simplify the definitions of $f^{\sigma}$ and $f^{\pi}$ as follows,
e.g. \cite{Distr} \cite{Expan}:
\begin{prop}
  $ $

  Let $\mathcal{K}$, $\mathcal{L}$ be bounded distributive lattices and $f : \mathcal{K} \to \mathcal{L}$. Then:
  \begin{enumerate}
    \item $f^{\pi}(u) = \bigwedge \{ \bigvee \{ f(a) \: | \: a \in \mathcal{K}, a \leq y \} \: | \: u \leq y \in \mathcal{I}(\mathcal{K}^{\sigma}) \}$, where $u \in \mathcal{K}^{\sigma}$
    \item $f^{\sigma}(u) = \bigvee \{ \bigwedge \{ f(a) \: | \: a \in \mathcal{K}, x \leq a \} \: | \: u \geq x \in \mathcal{F}(\mathcal{K}^{\sigma}) \}$, where $u \in \mathcal{K}^{\sigma}$
    \item $f^{\sigma}(x) = f^{\pi}(x) = \bigwedge \{ f(a) \: | \: x \leq a \in \mathcal{K} \}$, where $x \in \mathcal{F}(\mathcal{K}^{\sigma})$
    \item $f^{\sigma}(y) = f^{\pi}(y) = \bigvee \{ f(a) \: | \: a \in \mathcal{K}, a \leq y \}$, where $y \in \mathcal{I}(\mathcal{K}^{\sigma})$
  \end{enumerate}
\end{prop}
As a consequence, for each $u \in \mathcal{K}^{\sigma}$ one has $f^{\sigma}(u) = \bigvee \{ f^{\sigma}(x) \: | \: u \geq x \in \mathcal{F}(\mathcal{K}^{\sigma}) \}$.
The third and fourth items denote that $f^{\sigma}$ and $f^{\pi}$ send filter (ideal) elements to filter (ideal) ones. A map $f : \mathcal{K} \to \mathcal{L}$ is called \emph{smooth}, if $f^{\pi}(u) = f^{\sigma}(u)$ for each $u \in \mathcal{K}^{\sigma}$. In particular, a map is smooth if it preserves or reverses finite joins or meets.
For instance, modal operators $\Box$ and $\Diamond$ are smooth and their smoothness since $\Box$ and $\Diamond$ are meet and join hemimorphisms correspondingly. Thus, $\Box^{\sigma} = \Box^{\pi}$ and $\Diamond^{\sigma} = \Diamond^{\pi}$.

\bibliographystyle{aiml20}

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text:
Charles f.Porter
Charles Porter has succeeded where many other newcomers have failed. He has worked his way into Hollywood, managing to secure increasingly juicy roles while avoiding becoming an archetypal typecast.
He was born Charles Frederick Porter in Oxford, England on March 13, 1979 to a British mother and American father, and saw his destiny in the National Football League (USA) until a back injury sidelined his dreams at the age of 23.
Charles lived in Oxford until age 4, when his family migrated to the United States. His parents later split, and Charles went on to have an uncomplicated middle-class childhood, attending highly credited prep schools in the North East. He excelled in his studies, and became fluent in French as a teenager. But it was in sports that his greatest passion lied, and as a boy, he dreamed of becoming a professional athlete.
Charles' athletic talents eventually got him in the top 100 recruits out of high school, earning him a scholarship to several colleges across the country. Porter chose to pursue football at Duke University (North Carolina) for the school's academic standards and starred as defensive end under his position coach, Scott Brown, someone who would become a mentor to Charles, instilling a 'never quit' mentality into the young star's life.
Charles studied cultural anthropology and surfaced as a vicious college football player, winning Teammate of the year award with most valuable honors; six sacks, leading his team in tackles for a loss. He graduated with a Bachelor of Arts in Cultural Anthropology.
Perhaps one of the highlights of Charles' career was playing against the national champions Florida State Seminoles in front of a sellout crowd of 97,000 at Doak Campbell Stadium in Tallahassee, Florida.
While several of his opponents went on to play in the National Football League, a major injury to his lower back would seize Charles' dream, and turn his attention to an inner ambition.
Sixty-five pounds lighter, Charles settled in California to pursue a career in entertainment, it was here that he caught the attention of Executive Producer, Ben Katz.
Shortly after moving to LA in October of 2002, Charles landed a spot as a contestant on ESPN's Beg, Borrow and Deal.
The show, which was originally called "Beg, Borrow and B.S.", featured two teams of four pitted against each other. In Europe, episodes were broadcast on the North American Sports Network in 2005 and in Canada on GameTV beginning in 2007.
Almost a year would pass before Charles would land his first television role, and it came in the form of a teenage heart throb name Eric, on the hit sitcom The Parkers, a spin-off of Brandy's sitcom 'Moesha.'
Later the same year, Charles' career turned fashion, when he was cast in a touring runway show, The Ebony Fashion Fair (featuring up and coming as well as established Black Fashion Designers). It was here that producer Marty Casey (A House Divided); noticed, and cast Charles as a lead in the show playing in the Scottish Richt Theatre, St. Louis, Missouri. While being asked to go on tour with the show, Charles chose to appear in three shows, performing in front of 1500 people each night.
Charles continued to struggle through commercial parts and small roles until the summer of 2005, when he auditioned, and landed, a starring role in writer/director H. M. Coakley's urban horror-comedy, Holla. The film was produced by Coakley's wife and business partner, Camille Irons Coakley under their Rockstone Pictures banner and distributed by Lionsgate Films.
That same year, Adidas scouted the 6 foot 3" x-college football star in a two-year international campaign; featuring Charles in casual luxury to promote the apparel. Charles' photos ranged from dark and edgy to sensually appealing, displaying a combination of elegance and masculinity to the lifestyle brand.
While continuing to model and search for the right acting gigs, Charles had a chance encounter with executive producer Ben Katz while moving into his old apartment. The meeting in the West Hollywood complex lead to Katz casting Charles in a drama he was producing called, Johnny Was. Charles' dual citizenship allowed him to fly to Belfast, Ireland, and fit into the role of a British henchman named Digs.
While shooting the film in the summer of 2005, Charles produced and starred in another Ben Katz production, the psycho-thriller, Nailed.
Premiering in Cannes, and receiving much critical acclaim throughout its release, the project was distributed by Image Entertainment and proved to be one of the highlights of Charles' film career.
In a departure from film, Charles returned to the United States, guest starring on The CBS legal drama, Close to Home as a police officer. He went on to guest star on House MD as a street operator, his face opening up the medical drama to some 40 million viewers. The role opened news doors for Charles, including a national campaign with American Airlines.
Just one month later, Charles landed a guest starring role on one of the biggest series in America, the CBS drama, Without A Trace, and followed up with an appearance on Fox's dark -amusing drama BONES (ironically inspired by real-life forensic anthropologist Kathy Reichs).
Most recently, Charles was cast in the LA based Cabaret, Harlem's Night, the 1940's themed bed time story. The young star played Joe, and hit the stage singing, dancing and acting in his guest role.
Now just 28 years old, Charles Porter appears to be on the fast track to stardom. What is interesting now is what the future will bring. His captivating screen presence and versatile performances could allow him to become this generation's Sydney Poitier. His fun nature, sense of humor and classic good looks could just as easily allow him to mature into a sort of Marlon Brando. One thing is certain, expect him to be labeled as one of the most exciting new actors to be detected by the Hollywood radar.
Nina Ameri • (310) 300-2920 • nameri@agmblaw.com
© 2021 Charles F. Porter. Don't steal. It's not nice.

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What Are the Benefits of Buying A Home Before the End of The Year?
Are you planning on buying a home but aren't sure if you should plan on buying one now or wait until next year?
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To get started with buying a home in Southern California contact the Fred Sed Group today by calling us at (800) 921-9231 or connect with us online.


text:
Graduate Assistants & Teaching Assistants
PSRC - Psychological Services and Research Centre
Colloquium Talk - Dr. Ada L. Sinacore
Thursday, November 10, 2022 - 04:15 to 05:30
The Colloquium Committee would like to invite you to the colloquium presentation on Thursday, November 10th at 4:15pm via Zoom.
Zoom link for the colloquium.
Ada L. Sinacore, Ph.D. is an associate professor in the Counselling Psychology Program in the Faculty of Education at McGill University.
As well, they just completed their term as program director and are an associate member of the Institute for Gender, Sexuality, and Feminist Studies, and the Director of the Social Justice and Diversity Research Lab. Additionally, Professor Sinacore is the current past-president of the Canadian Psychological Association and sits on the board of the International Council of Psychologist. Professor Sinacore is a member of the advisory council of the Global Network of Psychologists for Human Rights; the Psychology Coalition at the United Nations; and is co-Chairing Psychology Day at the United Nations.
Professor Sinacore has over 25 years of experience working in the U.S., Canada, and abroad, and is internationally recognized for their expertise and extensive presentations, publications and activism addressing social justice and human rights concerns at the individual, institutional, societal, governmental, and policy levels. They are actively involved in research that addresses topics related to societal inequities and disparities in education and mental health. Professor Sinacore has received numerous honors and awards including, but not exhaustively, the Oliva Espin Award for Social Justice Concerns in Feminist Psychology, from the Association for Women in Psychology. They are a Fellow of the Canadian Psychological Association and received the Distinguished Member Award from the Section on Counselling Psychology (CPA).
At McGill University, Dr. Sinacore received the Faculty of Education Award for Distinguished Teaching and is recognized as a Distinguished Supervisor. Recently, Dr. Sinacore received the McGill University, Award for Equity and Community Building. Internationally, Dr. Sinacore was a Distinguished Visiting Scholar with the Department of International Cooperation and Science Education, in Taiwan and has been a visiting scholar in a number of universities around the world. In 2019, she collaborated with the Canadian Embassy in Chile to work with university students, women leaders, and NGOs to discuss ways to address sexual violence in their respective institutions. Professor Sinacore has been a key speaker at the XV and VII International Weeks at the University of Chile.
Download the colloquium announcement flyer. (PDF format)


For the best seat in the house, you can't beat a state-of-the-art TV center.
Here are the DIY Basics for creating a family TV center.
Paint walls a dark flat color so they won't reflect light.
Hang blackout shades on windows.
Install theater-style wall sconces, which focus light on wall, not on TV screen.
Get battery-powered fixtures and you won't have to run wires.
For comfortable viewing, mount TV with middle of screen at eye level when you're seated, 36 to 40 inches, and remember you may recline while watching TV.
Hang at least four surround-sound speakers, one on either side of TV about 30 degrees off center and two more on pivoting mounts on side walls at ends of sofa.
Standard height for speakers is ear level when you're standing; this prevents blast of sound in ears when you're seated.
Tip: A mounting kit will make hanging the TV easier.
Tip: Get wireless speakers and you won't have to drill through walls.
To access shows, movies, and music anytime get whole-home HD DVR system with large storage capacity. Have system professionally installed.
Get auxiliary units so you can watch shows from other rooms.
Stock mini fridge with snacks and drinks.
Make the most of your home's media room experience with the addition of a wall or ceiling-mounted projector screen.
Build room-darkening window panels for your home theater. Made from solid-wood, the panels are hung on a wall-mounted track system so they easily open and close. A decorative design carved into the panels allows them to double as artwork.


We've reported extensively here on allegations of cronyism and other problems with the Department of Housing and Urban Development's work in post-Katrina New Orleans -- problems that culminated in the resignation earlier this year of former HUD Secretary Alphonso Jackson, who remains under criminal investigation for his role in questionable contracts in New Orleans, Philadelphia and the Virgin Islands.
The contract awards that staff members questioned took place within programs, heavily promoted by Jackson, to help small, minority-owned businesses get a bigger share of the roughly $1 billion in public contracts HUD awards each year. During Jackson's tenure, the proportion of contracts awarded to small black- and Hispanic-owned businesses, including under the Section 8(a) program, rose from 6 percent to nearly 35 percent. The proportion of contracts open to full competition decreased from 71 percent to 33 percent, federal records show.
* Harrington, Moran and Barksdale Inc. (HMBI), an Arlington, Texas-based company that's led by two former Reagan administration officials and has close relations with senior Republican appointees at HUD. In 2004, the company -- which had previously managed only a handful of apartment buildings and development projects -- landed $71 million in HUD contracts overseeing the upkeep and sale of defaulted homes.
* National Housing Group, a Miami-based property management firm that's contributed generously to GOP campaigns, won $50 million in contracts from 2003 to 2007. Its second-highest executive has been indicted for allegedly falsifying reimbursement requests to HUD.
* Drayton, Drayton & Lamar of Georgia had received about $1 million in HUD business through 2002. But since 2003, it landed $35 million in data-technology and information-management contracts at HUD despite concerns about its performance. "Its president has socialized with a HUD official," the Post noted.
It also appears that HUD brass retaliated against lower-level employees who voiced objections to the deals. For example, when veteran agency contracting specialist Gloria Freeman repeatedly raised concerns about HMBI, she was transferred to a policy job and later retired. And when Ed Girovasi -- a 33-year HUD employee -- was asked to review an NHG claim for $8 million, he concluded the firm had exaggerated and actually owed HUD $250,000. He too was transferred to a policy position and later retired.


text:
Tcaci is a Moldovan occupational surname, of Slavic origin, meaning "weaver". It may refer to:

 Leonid Tcaci (b. 1970), Moldovan football manager
 Svetlana Șepelev-Tcaci (b. 1969), Moldovan long-distance runner
 Zlata Tcaci (1928–2006), Moldovan composer

See also
 Tkach
 Tcaciuc

Romanian-language surnames
Surnames of Moldovan origin

redpajama_set_name: RedPajamaWikipedia


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In the Book of Luke, Jesus says, "Give, and it will be given to you. They will pour into your lap a good measure– pressed down, shaken together, and running over. For by your standard of measure it will be measured to you in return."
At CrossRoads Missions, giving back to the communities we serve is the cornerstone of what we do. In more than 26 years, our organization has helped thousands of people and allowed thousands more the opportunity to serve. In 2020, we would like to do so much more… but we can't without your help.
That's why we're asking you to please consider contributing to our year-end fundraising drive. The deadline is 11:59 P.M. EST on December 31, and all proceeds will be earmarked to our developmental fund, which manages central operations for CrossRoads.
Here's where your donation goes.
Our developmental fund oversees the following aspects of CRM:
-Enriching services provided in each field
-Logistical support for full-time personnel in each field
-In-person visits by CrossRoads staff to valuable donors, such as yourself
-Marketing, to continue to spread the word about CrossRoads and our mission trips
-Ministry planning
-Business planning
-Recruiting interns and full-time staff
By accomplishing our fundraising goal, CrossRoads will be able to expand and provide even more opportunities for service in 2020.
Remember that CrossRoads is a 501(c) non-profit, meaning all donations and contributions are tax-deductible. Click here to make your donation.
On behalf of everyone at CrossRoads, we'd like to wish you a happy and blessed New Year!


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Proactive Measures to Reduce Risk of Healthcare Data Breach
Posted on 11-December-2017 by Rajeev Rajagopal
Healthcare security breach is widespread and reports indicate that an average of one healthcare organization experiences a data breach every day. Medical transcription outsourcing to a reliable service provider can ensure HIPAA compliant medical documentation practices. However, electronic health records (EHRs) are virtually exist everywhere, making them an easy target for hackers. Industry experts stress that healthcare organizations need to take a proactive approach to safeguarding EHRs, reducing risk of security breach, protecting their reputation, and reducing risk of liability claims.
In 2015, more than 113 million persons were affected by a data breach. Ponemon Institute reports that the average cost of one stolen or lost record is $363. The US Department of Health and Human Services Office for Civil Rights (OCR) received reports of as many as 264 data breaches affecting at least 4.4 million patients as of November 21, 2017.These breaches include hacking/IT incidents, unauthorized access/disclosure, theft, improper disposal and loss. The breaches affected physician and dental practices, hospitals, insurers, medical equipment suppliers and health systems. Rather than just relying on detection software and hardware, healthcare entities have to understand their risks and vulnerabilities, and then take appropriate measures to manage them.
A recent Beckers ASC Review report recommends risk assessment and other steps to reduce risk of data breach in ambulatory surgery centers (ASCs):
Data breach risk analysis: Performing a risk assessment could identify gaps or vulnerabilities that can be exploited to access to PHI, according to HIPAA Breach News (HBN). System weaknesses can expose an organization to liability for breach of confidentiality and invasions of privacy. Inappropriate uses or disclosures of information can cause negative publicity and make patients to choose other providers. IT-related errors and loopholes can corrupt or destroy vital data, or result in inappropriate alteration or manipulation of data.
Education and training: All persons in the organization should be trained on security and privacy rules that apply to healthcare and the impact that they can have. Participation in cyber protection means understanding what's happening in the wider security landscape. All staff should know why PHI needs to be protected and also informed about the importance of observing and reporting suspicious events. Regular training sessions and annual reorientation can keep staff up to date on these matters.
Policies specific to ASCs: The Beckers ASC Review report says that ASCs need to implement certain policies that are specific to their needs:
High-profile patients: ASCs need to take special care to keep records of high-profile patients private and confidential. This can be done by assigning an alias within the EHR for them, and restricting access to the "need to know" list. At the same time, exemptions will be needed to allow internal access for treatment, administrative or other specific purposes. Frequent audits should be conducted to identify who is accessing these records and whether they have a legitimate reason to do so.
Bring Your Own Device (BYOD) policy: BYOD policy should be properly defined. The best policy would be to include only organization administered devices. If physicians, nurses and administrative staff are allowed to access PHI on personal laptops, tablets and smartphones, the parameters should be defined. Requirements for device encryption should be spelt out. Phones can also be issued internally to staff members, an expensive but worthwhile investment option to protect PHI.
Secure storage, destruction and disposal of PHI: There should be clearly defined policies for retention and destruction of PHI in paper and electronic format. Data backup is also crucial.
Encryption: Encryption is essential to protect PHI. The data encryption policies of medical transcription companies are a good example. Data encryption ensures that data is protected at all times.
Physical security: This includes the use of alarm systems to prevent break-ins and access to computers via an unlocked office. Care must be taken to ensure that paper records are not stored in an area with unrestricted access, especially in buildings with space constraints.
Define staff responsibilities: Privacy and security job descriptions must clearly spell out who is responsible for privacy and security.
It is important to keep online traffic concealed with block tracking cookies. This will limit the ability of third parties to follow online traffic and prevent unauthorized persons from accessing secure accounts. Performing regular audits is an important best practice when it comes to security. Security audits should be done annually. Failure to perform HIPAA-compliant security analyses can attract penalties starting in the $100,000 range.
Most healthcare providers partner with a third-party vendor for medical billing and coding support as well as medical transcription services. It is critical to choose a vendor whose security measures meet the highest standards. As incidents of PHI breach increase, only a proactive approach can safeguard patient data and reduce the risk of cyber attacks.
EHR Transcription	| Healthcare Data Breach, Medical Transcription Outsourcing, Medical Transcription Services


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Q: What does it mean that DX12 has heterogeneous multi GPU support and Vulkan doesn't? I've been looking into using vulkan to leverage compute capabilities on integrated graphics and discrete graphics capabilities.  However, according to this article, despite having multi gpu support through compatible SLI configurations, Vulkan does not have heterogeneous multi gpu support.  The same article goes on to claim the DX12 does in fact have this kind of support, going "a step further".
In my application I want to take advantage of the compute capabilities of two separate devices which do not have SLI/Xfire compatibility, but are also not necessarily working on the same compute workload, memory need not be shared between each to compute their workloads.
Is it not possible for me to simply have two devices do two separate things in the same instance, and if necessary, share memory via copying to host first? This answer appears to imply that I can actually do that, so I'm confused as to what heterogeneous multi gpu actually means for DX12, and how that is different from what I can do in Vulkan.

redpajama_set_name: RedPajamaStackExchange


Tickets are available for the music and registration is now open for the Throwdown!
Don't miss a bit of it!
Visit BigOrangeMusicFestival.com for details, tickets, full music line-up, and schedule of events, and follow them on Facebook for updates.


The creek is in better shape today than fifty years ago.
Then, creek banks were crowded with brush. Now, you can stand on the bank and easily cast your tackle. There are still cat tails in the creek but they are controlled by a local wildlife biologist for a monthly stipend.
Fifty years ago there were perch in the water, small fish that strike impulsively, put up a fight, and have lots of bones to work around at the dinner table. We ate them fried in a blanket of corn meal along with cornbread, black eyed peas and Texas toast fixed by Grandma. In the creek, we kids waded in undershorts seining for minnows to use as bait. For city kids, the creek and the ranch were a place to look forward to visiting when school shut down for the summer.
The water today is dark, opaque, ten foot deep in the middle. It's surface is a mirror reflecting trees on the other side of the bank. Like so much of nature, you can feel a lot more beneath the surface than you can see.
Growing up, I had no idea I would be fishing the creek when I got old.
Even the future can't escape the past.
Hey Scott – I am happy that the creek photos made it to your Trek's page. Hope you and Alan caught some fish!
Always good to revisit the ranch. Your Grandfather and our Aunt made it a great place to be in the summer. They were very kind. Lots of memories on the creek. We didn't catch much but it is still peaceful around the water. Thanks for stopping by Scotttreks.


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Justin Domke and Yiannis Aloimonos's paper, A probabilistic notion of camera geometry", received a best paper prize at the 2006 Photogrammetric Computer Vision Conference in Bonn, Germany.
Justin Domke and Yiannis Aloimonos 's paper, " A probabilistic notion of camera geometry ", received a best paper prize at the 2006 Photogrammetric Computer Vision Conference in Bonn, Germany. read more
V.S. Subrahmanian was a winner of a 2006 Computerworld Horizon Award for the Opinion Analysis System software developedhere at Maryland in partnership with the University of Naples in Italy.
V.S. Subrahmanian was a winner of a 2006 Computerworld Horizon Award for the Opinion Analysis System software developedhere at Maryland in partnership with the University of Naples in Italy. Article read more
Jonathan Katz was invited to give a lecture in the Cryptography and Security" session at the 2006 SIAM Conference on Discrete Mathematics.
Jonathan Katz was invited to give a lecture in the "Cryptography and Security" session at the 2006 SIAM Conference on Discrete Mathematics . read more
Chris Conroy was cited in a Wall Street Journal article on ID card security. Colleges Face Privacy Issues As ID Cards Collect More Data
Chris Conroy was cited in a Wall Street Journal article on ID card security. Colleges Face Privacy Issues As ID Cards Collect More Data read more
Aditya Kalyanpur, advisor Jim Hendler, was the primary author for the paper, Repairing Unsatisfiable Concepts in OWL Ontologies" which was co-selected as best paper for the 3rd Annual European Semantic Web Conference.
Aditya Kalyanpur , advisor Jim Hendler , was the primary author for the paper, " Repairing Unsatisfiable Concepts in OWL Ontologies " which was co-selected as best paper for the 3rd Annual European Semantic Web Conference. read more
Jandelyn Plane received the CMPS Thelma M. Williams Advisor of the Year Award for 2006
Jandelyn Plane received the CMPS Thelma M. Williams Advisor of the Year Award for 2006 read more
Michelle Hugue received the CMPS Outstanding Instructor Award for 2006
Michelle Hugue received the CMPS Outstanding Instructor Award for 2006 read more
John Shin, a senior CS major, is the college's Outstanding Undergraduate for 2006, and will be the student speaker at the college's graduation ceremony.
John Shin, a senior CS major, is the college's Outstanding Undergraduate for 2006, and will be the student speaker at the college's graduation ceremony. View the transcript of Shin's Graduation Speech . read more
Julian Mestre was awarded a Dean's Dissertation Fellowship for Fall 2006
Julian Mestre was awarded a Dean's Dissertation Fellowship for Fall 2006 read more
Pete Stewart has been elected a Distinguished University Professor. He is the first person in our department to be awarded this honor by the campus.
Pete Stewart has been elected a Distinguished University Professor. He is the first person in our department to be awarded this honor by the campus. read more