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Concept
Presentation of a monoid
Summary
In algebra, a presentation of a monoid (or a presentation of a semigroup) is a description of a monoid (or a semigroup) in terms of a set Σ of generators and a set of relations on the free monoid Σ∗ (or the free semigroup Σ+) generated by Σ. The monoid is then presented as the quotient of the free monoid (or the free semigroup) by these relations. This is an analogue of a group presentation in group theory.
As a mathematical structure, a monoid presentation is identical to a string rewriting system (also known as a semi-Thue system). Every monoid may be presented by a semi-Thue system (possibly over an infinite alphabet).
A presentation should not be confused with a representation.
The relations are given as a (finite) binary relation R on Σ∗. To form the quotient monoid, these relations are extended to monoid congruences as follows:
First, one takes the symmetric closure R ∪ R−1 of R. This is then extended to a symmetric relation E ⊂ Σ∗ × Σ∗ by defining x ~E y if and only if x = sut and y = svt for some strings u, v, s, t ∈ Σ∗ with (u,v) ∈ R ∪ R−1. Finally, one takes the reflexive and transitive closure of E, which then is a monoid congruence.
In the typical situation, the relation R is simply given as a set of equations, so that . Thus, for example,
is the equational presentation for the bicyclic monoid, and
is the plactic monoid of degree 2 (it has infinite order). Elements of this plactic monoid may be written as for integers i, j, k, as the relations show that ba commutes with both a and b.
Presentations of inverse monoids and semigroups can be defined in a similar way using a pair
where
is the free monoid with involution on , and
is a binary relation between words. We denote by (respectively ) the equivalence relation (respectively, the congruence) generated by T.
We use this pair of objects to define an inverse monoid
Let be the Wagner congruence on , we define the inverse monoid
presented by as
In the previous discussion, if we replace everywhere with we obtain a presentation (for an inverse semigroup) and an inverse semigroup presented by .
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