Clifford semigroups

Abbreviation: CliffSgrp

Definition

A Clifford semigroup is an inverse semigroup S = (S,·,⁻¹) that is also completely regular.

Definition

A Clifford 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 and xx⁻¹y⁻¹y = y⁻¹yxx⁻¹ and xx⁻¹ = x⁻¹x.

Morphisms

Let S and T be Clifford 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	No
Strong amalgamation property	No
Epimorphisms are surjective	Yes

Finite members

[Size 1]?: 1
[Size 2]?:
[Size 3]?:
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Subclasses

Groups

Superclasses

Completely regular semigroups
Inverse semigroups