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·y)·z = x·(y·z),
e is an identity for ·: e·x = x and x·e = x.
Let M and N be monoids. A morphism from M to N is a function h : M→N that is a homomorphism: h(x·y) = h(x)·h(y) and h(e) = e.
(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.
|Equational theory||decidable in polynomial time|
|Congruence extension property|
|Definable principal congruences|
|Equationally definable principal congruences||no|
|Strong amalgamation property||no|
|Epimorphisms are surjective||no|