For questions about elements which satisfy $x\cdot x=x$ where $\cdot$ is a composition law.
4
votes
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 ...
0
votes
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 $f$?
7
votes
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 = ...
1
vote
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 ...
3
votes
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 ...
0
votes
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 $x, y, z \in S$:
$f(x, y) ...
0
votes
1answer
78 views
Determine if the matrix is idempotent?
I am dealing with an example to show that the matrix($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 ...
2
votes
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 ...
0
votes
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 ...
5
votes
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 ...
