Full Lattice Convergence on Riesz Spaces
The full lattice convergence on a locally solid Riesz space is an abstraction of the topological, order, and relatively uniform convergences. We investigate four modifications of a full convergence $\mathbb{c}$ on a Riesz space. The first one produces a sequential convergence $\mathbb{sc}$. The second makes an absolute $\mathbb{c}$-convergence and generalizes the absolute weak convergence. The third modification makes an unbounded $\mathbb{c}$-convergence and generalizes various unbounded convergences recently studied in the literature. The last one is applicable whenever $\mathbb{c}$ is a full convergence on a commutative $l$-algebra and produces the multiplicative modification $\mathbb{mc}$ of $\mathbb{c}$. We study general properties of full lattice convergence with emphasis on universally complete Riesz spaces and on Archimedean $f$-algebras. The technique and results in this paper unify and extend those which were developed and obtained in recent literature on unbounded convergences.