Completely regular semigroups

Abbreviation: CompRSgrp

Definition

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

Morphisms

Let S and T be completely regular 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	No
Definable principal congruences
Equationally definable principal congruences	No
Amalgamation property
Strong amalgamation property
Epimorphisms are surjective

Finite members

[Size 1]?: 1
[Size 2]?:
[Size 3]?:
[Size 4]?:
[Size 5]?:
[Size 6]?:
[Size 7]?:

Subclasses

Clifford semigroups
Bands
[Commutative completely regular semigroups]?

Superclasses

Regular semigroups