[Home]Commutative semigroups

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Abbreviation: CSgrp

Definition

A commutative semigroup is a semigroup S = (S,·) such that · is commutative:   xy = yx.

Definition

A commutative semigroup is a structure S = (S,·), where · is an infix binary operation, called the semigroup product, such that
· is associative:   (xy)z = x(yz)and
· is commutative:   xy = yx.

Morphisms

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

Some results

Examples

(N, + ), the natural numbers, with additition.

Properties

Classtype variety
Equational theory decidable in polynomial time
Quasiequational theory decidable
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
Definable principal congruences
Equationally definable principal congruences no
Amalgamation property no
Strong amalgamation property no
Epimorphisms are surjective no

Finite members

Search for finite commutative semigroups

Size 1:  1
Size 2:  3
Size 3:  12
Size 4:  58
Size 5:  325
[Size 6]?:  2143
[Size 7]?:  17291

Subclasses

Semilattices
Commutative monoids

Superclasses

Semigroups
[Partial commutative semigroups]?


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Last edited May 29, 2003 10:05 am (diff)
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