presentation of inverse monoids and inverse semigroups
Let (X∐X-1)∗ be the free monoid
with involution on X, and T⊆(X∐X-1)∗×(X∐X-1)∗ be a binary relation between words. We denote by Te [resp. Tc] 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 Inv1⟨X|T⟩. Let ρX be the Wagner congruence on X, we define the inverse monoid Inv1⟨X|T⟩ presented by (X;T) as
Inv1⟨X|T⟩=(X∐X-1)∗/(T∪ρX)c. |
In the previous dicussion, if we replace everywhere (X∐X-1)∗ with (X∐X-1)+ we obtain a presentation (for an inverse semigroup) (X;T) and an inverse semigroup Inv⟨X|T⟩ 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
FIM(X)=Inv1⟨X|∅⟩=(X∐X-1)∗/ρX,[resp. FIS(X)=Inv⟨X|∅⟩=(X∐X-1)+/ρX]. |
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 |
---|---|
Canonical name | PresentationOfInverseMonoidsAndInverseSemigroups |
Date of creation | 2013-03-22 16:11:01 |
Last modified on | 2013-03-22 16:11:01 |
Owner | Mazzu (14365) |
Last modified by | Mazzu (14365) |
Numerical id | 10 |
Author | Mazzu (14365) |
Entry type | Definition |
Classification | msc 20M05 |
Classification | msc 20M18 |
Synonym | presentation |
Synonym | generators and relators |