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Monoids

Abbreviation: Mon

Definition

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

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

e is an identity for : ex=x, xe=x.

Morphisms

Let M and N be monoids. A morphism from M to N is a function h:MarrowN that is a homomorphism:

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

Examples

Example 1: XX,,idX, the collection of functions on a sets X, with composition, and identity map.

Example 1: M(V)n,,In, the collection of n×n matrices over a vector space V, with matrix multiplication and identity matrix.

Example 1: Σ,,λ, the collection of strings over a set Σ, with concatenation and the empty string. This is the free monoid generated by Σ.

Basic results

Properties

Finite members

f(1)=1f(2)=2f(3)=7f(4)=35f(5)=228f(6)=2237f(7)=31559

Subclasses

Superclasses

References


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