Let $S$ be a finite semigroup. For two idempotents $e, f$ we set $$ f \le e :\Leftrightarrow ef = f $$ Now let $s \in S$ and $e,f \in S$ be two idempotents. Suppose $s = se = sf$ and there exists $x, y\in S$ such that $$ s = sx, e = xy, […]

I have a set of matrices $A_1,\ldots,A_n$ and another set $B_i = A_i^{-1}$ for $i = 1,\ldots, n$ (I assume $A_i$ are invertible). Let $\mathcal{A} = \{A_i\} \cup \{B_i\}$. What are some simple conditions under which $$\prod_{k=1}^r C_k = I$$ (for $C_k \in \mathcal{A}$ and $r$ an integer) if and only if the $C_k$ are […]

Let $(S,\cdot)$ and $(T,\circ)$ be semigroups (or some algebraic structure with an operation), then they are anti-isomorphic if there exists some $\varphi : S \to T$ such that $$ \varphi(xy) = \varphi(y) \circ \varphi(x). $$ Now for what is this notion useful? The notion of isomorphism is useful as this basically says that isomorphic structures […]

As you see, I did a finite semigroup and then try to find its possible ideals in a very basic way (with the hand of a friend): > f:=FreeSemigroup(“a”,”b”);; a:=f.1;; b:=f.2;; s:=f/[[a^3,a],[b^2,a^2],[b*a,a*b^3]];; e:=Elements(s);; w1:=List([1..Size(e)],i->Elements(SemigroupIdealByGenerators(s,[e[i]])));; w2:=DuplicateFreeList(w1);; w3 := Combinations(w2);; for i in [1..Size(w3)] do w3[i] := DuplicateFreeList(Flat(w3[i]));; od;; w3 := DuplicateFreeList(w3); > [ [ ], [ […]

$\mathscr T_X$ will denote the set of all functions from a non-empty set $X$ into itself, with the binary operation of composition $\circ$ making it a semigroup, called the full transformation semigroup on $X$. Is there a topology on the set $\mathscr T_X$ such that $\circ:\mathscr T_X\times \mathscr T_X\longrightarrow \mathscr T_X$ is a continuous function […]

Suppose a semigroup (possibly infinite) presentation is given with generating set $S$ and relations $R$. I need to prove using Bergman’s diamond lemma that the semigroup is non-zero i.e, I have to give normal forms of elements of the semigroup. Suppose I could guess the set of irreducible elements and I have also an order […]

For each $n \in \mathbb{N}$, let $F_n$ be a finite set with $n$ elements. For any function $f : F_n \to F_n$ and $k \in \mathbb{N}$, $f^k$ is the result of composing $f$ with itself $k$ times. Say that $n$ distinguishes powers $i$ and $j$ iff there is some function $f : F_n \to F_n$ […]

I have been looking at this for hours now. Why is it true that idempotents of an inverse semigroup commute? It seems like this should be straightforward but I just can’t get it. Any help is greatly appreciated.

I did Google search and can’t find a good answer. I thought I should ask experts here. Cayley graph is defined for groups. My question is: Is there a special name for the Cayley graph of semigroups?

Let $S$ be a semigroup. Show that $S$ is a group if and only if $aS=S=Sa$ for all $a\in S$. Since it is if and only statement, we have to show that if $S$ is a group then $ aS=S=Sa$, which I already know how to do. The other part: if you have $aS=S=Sa$, then […]

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