[Home]Cancellative commutative monoids

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Abbreviation: CanCMon

Definition

A cancellative commutative monoid is a cancellative monoid M = (M,·,e) such that · is commutative:   x·y = y·x.

Morphisms

Let M and N be cancellative commutative monoids. A morphism from M to N is a function h : MN that is a homomorphism: h(x·y) = h(xh(y)  and  h(e) = e.

Some results

All commutative free monoids are cancellative.
All finite commutative (left or right) cancellative monoids are reducts of abelian groups.

Examples

(N, + ,0), the natural numbers, with addition and zero.

Properties

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

Finite members

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

Subclasses

Abelian groups
[Cancellative commutative residuated lattices]?

Superclasses

Cancellative commutative semigroups
Cancellative monoids
Commutative monoids


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Last edited March 25, 2003 9:32 am (diff)
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