# presentation of inverse monoids and inverse semigroups

Let $\left(X\amalg X^{-1}\right)^{\ast}$ be the free monoid with involution on $X$, and $T\subseteq\left(X\amalg X^{-1}\right)^{\ast}\times\left(X\amalg X^{-1}\right)^% {\ast}$ be a binary relation between words. We denote by $T^{\mathrm{e}}$ [resp. $T^{\mathrm{c}}$] the equivalence relation [resp. congruence] generated by $T$.

A presentation (for an inverse monoid) is a couple $(X;T)$. We use this couple of objects to define an inverse monoid $\mathrm{Inv}^{1}\left\langle X|T\right\rangle$. Let $\rho_{X}$ be the Wagner congruence on $X$, we define the inverse monoid $\mathrm{Inv}^{1}\left\langle X|T\right\rangle$ presented by $(X;T)$ as

 $\mathrm{Inv}^{1}\left\langle X|T\right\rangle=\left(X\amalg X^{-1}\right)^{% \ast}/(T\cup\rho_{X})^{\mathrm{c}}.$

In the previous dicussion, if we replace everywhere $\left(X\amalg X^{-1}\right)^{\ast}$ with $\left(X\amalg X^{-1}\right)^{+}$ we obtain a presentation (for an inverse semigroup) $(X;T)$ and an inverse semigroup $\mathrm{Inv}\left\langle X|T\right\rangle$ 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 $\mathrm{FIM}(X)$ [resp. $\mathrm{FIS}(X)$] and is defined by

 $\mathrm{FIM}(X)=\mathrm{Inv}^{1}\left\langle X|\varnothing\right\rangle=\left(% X\amalg X^{-1}\right)^{\ast}/\rho_{X},\ \ \mbox{[resp. \mathrm{FIS}(X)=% \mathrm{Inv}\left\langle X|\varnothing\right\rangle=\left(X\amalg X^{-1}\right% )^{+}/\rho_{X}]}.$

## References

• 1 N. Petrich, Inverse Semigroups, Wiley, New York, 1984.
• 2 J.B. Stephen, Presentation of inverse monoids, J. Pure Appl. Algebra 63 (1990) 81-112.
Title presentation of inverse monoids and inverse semigroups PresentationOfInverseMonoidsAndInverseSemigroups 2013-03-22 16:11:01 2013-03-22 16:11:01 Mazzu (14365) Mazzu (14365) 10 Mazzu (14365) Definition msc 20M05 msc 20M18 presentation generators and relators