Commutative monoids

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Definition

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

Definition

A commutative monoid is a structure M = (M,·,e), where · is an infix binary operation, called the monoid product, and e is a constant (nullary operation), called the identity element, such that
· is commutative:   x·y = y·x,
· is associative:   (x·yz = x·(y·z),
e is an identity for ·:   e·x = x.

Morphisms

Let M and N be 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.

Examples

(N, + ,0), the natural numbers, with addition and zero. The finitely generated free commutative monoids are direct products of this one.

Properties

 Classtype variety Equational theory decidable Quasiequational theory decidable First-order theory undecidable Locally finite no Residual size unbounded 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 Strong amalgamation property Epimorphisms are surjective

Size 1:  1
Size 2:  2
Size 3:  5
Size 4:  19
Size 5:  78
[Size 6]?:  421
[Size 7]?:  2637

Subclasses

Abelian groups
Semilattices with identity

Superclasses

Commutative semigroups
Monoids

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