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presentation of inverse monoids and inverse semigroups (Definition)

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 \vert T \right\rangle $. Let $ \rho_X$ be the Wagner congruence on $ X$, we define the inverse monoid $ \mathrm{Inv}^1\left\langle X \vert T \right\rangle $ presented by $ (X;T)$ as

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

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 \vert 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

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

Bibliography

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.



"presentation of inverse monoids and inverse semigroups" is owned by Mazzu.
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Other names:  presentation, generators and relators
Keywords:  Inverse Semigroups, Word Problem, Isomorphism Problem
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Cross-references: free inverse semigroup, free inverse monoid, inverse semigroup, Wagner congruence, objects, monoid, inverse, congruence generated by, equivalence relation, words, binary relation, free monoid with involution
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This is version 7 of presentation of inverse monoids and inverse semigroups, born on 2006-08-21, modified 2006-08-24.
Object id is 8271, canonical name is PresentationOfInverseMonoidsAndInverseSemigroups.
Accessed 1359 times total.

Classification:
AMS MSC: 20M18 (Group theory and generalizations :: Semigroups :: Inverse semigroups)
 20M05 (Group theory and generalizations :: Semigroups :: Free semigroups, generators and relations, word problems)
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