Boolean Inverse Semigroup

Inverse semigroup #

Let \((S,\cdot)\) be a semigroup¹. We say that \(S\) is an inverse semigroup if every element of \(S\) has a unique inverse, say, \(a^*.\)

\[ \begin{aligned} S&\longrightarrow S\\ a&\longmapsto a^*\text{ (or } a^{-1})\\ \text{s.t. }& aa^*a=a,\; a^*aa^*=a^*\\ \text{i.e. }& (a^*)^*=a,\; (ab)^*=b^*a^* \end{aligned} \]

and moreover \(a^*a\) & \(aa^*\) are idempotents,

\[ \begin{aligned} (a^*a)^2=(a^*a)(a^*a)=a^*(aa^*a)=a^*a\\ (aa^*)^2=(aa^*)(aa^*)=(aa^*a)a^*=aa^* \end{aligned} \]

and in fact, all the idempotents of \(S\) are in

\[E(S)=\left\{aa^*\mid a\in S\right\}=\left\{a^*a\mid a\in S\right\}\]

which is a commutative semigroup, called the commutative idempotent semigroup. If we define \(e\land f=ef\) then \(E(S)\) becomes a semilattice. Thus we refer to \(E(S)\) as the semilattice of idempotents of \(S.\)

Example. Let \(X\) be a set. We know that \(\mathrm{Sym}(X),\) the set of all bijections on \(X,\) forms a group. Instead of bijections, consider partial bijections \(f:A\to B,\) \(g:C\to D\) (with \(A,B,C,D\subset X\))

Notes & References #