# Monoids

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### Definition

A 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 associative:   (x·yz = x·(y·z),
e is an identity for ·:   e·x = x  and  x·e = x.

### Morphisms

Let M and N be 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

(XX,o,idX), the collection of functions on a sets X, with composition, and identity map.

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

(Σ*,·,λ), the collection of strings over a set Σ, with concatenation and the empty string. This is the free monoid generated by Σ.

### Properties

 Classtype Variety Equational theory decidable in polynomial time Quasiequational theory undecidable 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 no Strong amalgamation property no Epimorphisms are surjective no

### Finite members

Size 1:  1
Size 2:  2
Size 3:  7
Size 4:  35
Size 5:  228
[Size 6]?:  2237
[Size 7]?:  31559

### Subclasses

Cancellative monoids
Commutative monoids

### Superclasses

Semigroups
[Partial monoids]?

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