Commutative monoids

Abbreviation: CMon

Definition

A \emph{commutative monoid} is a monoids M=M,,eM=M,,e such that

is commutative: xy=yxxy=yx

Definition

A \emph{commutative monoid} is a structure M=M,,eM=M,,e, where is an infix binary operation, called the \emph{monoid product}, and ee is a constant (nullary operation), called the \emph{identity element}, such that

is commutative: xy=yxxy=yx

is associative: (xy)z=x(yz)(xy)z=x(yz)

ee is an identity for : ex=xex=x

Morphisms

Let MM and NN be commutative monoids. A morphism from MM to NN is a function h:MNh:MN that is a homomorphism:

h(xy)=h(x)h(y)h(xy)=h(x)h(y), h(e)=eh(e)=e

Examples

Example 1: N,+,0, the natural numbers, with addition and zero. The finitely generated free commutative monoids are direct products of this one.

Basic results

Properties

Finite members

f(1)=1f(2)=2f(3)=5f(4)=19f(5)=78f(6)=421f(7)=2637

Subclasses

Superclasses

References


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