Presentation of inverse semigroups and inverse monoid

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In mathematics, in the subfield of abstract algebra, a presentation for an inverse monoid is a pair

(X; T)

where

$(X\cup X^{-1})^*$

is the free monoid with involution on $X$ , and

$T\subseteq (X\cup X^{-1})^*\times (X\cup X^{-1})^*$

is a binary relation between words. We denote by

T e

[resp.

T c

]

the equivalence relation (respectively, the congruence) generated by $T$ .

We use this pair of objects to define an inverse monoid

$\mathrm{Inv}^1 \langle X | T\rangle$ .

Let $ρ X$ be the Wagner congruence on $X$ , we define the inverse monoid

$\mathrm{Inv}^1 \langle X | T\rangle$

presented by $(X; T)$ as

$\mathrm{Inv}^1 \langle X | T\rangle=(X\cup X^{-1})^*/(T\cup\rho_X)^{\mathrm{c}}.$

In the previous discussion, if we replace everywhere $({X\cup X^{-1}})^*$ with $({X\cup X^{-1}})^+$ we obtain a presentation (for an inverse semigroup) $(X; T)$ and an inverse semigroup $\mathrm{Inv}\langle X | T\rangle$ presented by $(X; T)$ .

A trivial but important example is the Free Inverse Monoid [resp. Free Inverse Semigroup] on $X$ , that is usually denoted by $FIM(X)$ [resp. $FIS(X)$ ] and is defined by

$\mathrm{FIM}(X)=\mathrm{Inv}^1 \langle X | \varnothing\rangle=({X\cup X^{-1}})^*/\rho_X,$