Commutative inverse semigroups

Abbreviation: CInvSgrp

Definition

A commutative inverse semigroup is an inverse semigroup S = (S,·,⁻¹) such that · is commutative: xy = yx.

Definition

A commutative inverse semigroup is a structure S = (S,·,⁻¹) such that
· is associative: (xy)z = x(yz),
· is commutative: xy = yx and
⁻¹ is an inverse: xx⁻¹x = x and (x⁻¹)⁻¹ = x.

Morphisms

Let S and T be commutative inverse semigroups. A morphism from S to T is a function h : S→T that is a homomorphism: h(xy) = h(x)h(y) and h(x⁻¹) = h(x)⁻¹.

Some results

Examples

Properties

Classtype	Variety
Equational theory
Quasiequational theory
First-order theory
Locally finite	No
Residual size
Congruence distributive	No
Congruence modular	No
Congruence n-permutable	No
Congruence regular	No
Congruence uniform	No
Congruence extension property	Yes
Definable principal congruences
Equationally definable principal congruences	No
Amalgamation property	Yes
Strong amalgamation property	Yes
Epimorphisms are surjective	Yes

Finite members

[Size 1]?: 1
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Subclasses

Abelian groups
Semilattices

Superclasses

Inverse semigroups