# Tagged Questions

For questions about elements which satisfy $x\cdot x=x$ where $\cdot$ is a composition law.

2answers
44 views

### Idempotents in a Quotient Ring

Let $R=\mathbb{Z}_p[x]/(x^p-x)$. Show that $R$ has exactly $2^p$ elements satisfying $r^2=r$. I know that for $f,g\in\mathbb{Z}_p[x]$, we have $f-g\in(x^p-x)$ if and only if $f(a)=g(a)$ for all ...
1answer
30 views

### Is there a name for a relationship like idempotence between two functions?

If $f(f(x)) = f(x) \quad \forall \space x$ then $f$ is idempotent. If $g(f(x)) = f(x) \quad \forall \space x$ then is there a term to describe the relationship between $g$ and $f$?
1answer
95 views

### Sum of idempotent matrices is Identity

[Ciarlet, Problem $1.1-10$] Let $A_k$, $1 \leq k\leq m$, be matrices of order $n$ satisfaying $$\sum_{k=1}^mA_k\ =\ I.$$ Show that the following conditions are equivalent. \$A_k = ...
1answer
44 views

### Formal Notation of Functions

Say i have a function $f$ with the following signature: $f: S \times S \to S$ and I wanted to formally represent that this function was idempotent and reflexive. How would i do right this? For ...
0answers
51 views

### Representation for idempotent semiring

I have a semi-ring whose multiplication is non-commutative, and addition is idempotent. That is, $ab \neq ba$ and $a + a = a$. The semi-ring is freely generated from a finite set $\Sigma$, the ...
0answers
88 views

### A special idempotent function

Usually an idempotent function is $f: S \rightarrow S$ such that $f \circ f = f$. But I need something slightly different: $f: S \times S \rightarrow S$, such that: For $x, y, z \in S$: \$f(x, y) ...
1answer
78 views

### Determine if the matrix is idempotent?

I am dealing with an example to show that the matrix($M = I − X(X'X)^{−1}X'$) is idempotent. X is a matrix with T rows and k columns and I the unit matrix of dimension T. And then to determine the ...
0answers
32 views

### Left continuous magmas with no fixed points

Let $X$ be a compact Hausdorff topological space, and $*: X^2\rightarrow X$ an associative map (so that $(X, *)$ is a semigroup) which is left continuous (for all $s\in X$, the map $t\mapsto ts$ is ...
0answers
32 views

### fully idempotent and co-idempotent

I want the proof of : For each prime number p, the sub module N = Zp ⊕0 of the Z-module M = Zp⊕Zp is not idempotent but N = Hom z (M;N)N. Z4 is a multiplication Z-module which is not fully ...
1answer
95 views

### Can a sum of idempotents vanish?

Let $A$ be a finite dimensional $\mathbb C$-algebra. Let $e_1,\ldots,e_r\in A$ be nonzero idempotents (with $r>0$), i.e. $e_i^2=e_i$. My question is: Can it happen that $e_1+\cdots+e_r=0$? I can't ...
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