A semigroup is a structure S = (S,·), where · is an infix binary operation, called the
semigroup product, such that
· is associative: (xy)z = x(yz).
Let S and T be semigroups. A morphism from S to T is a function h : S→T that is a homomorphism: h(xy) = h(x)h(y).
(XX,o), the collection of functions on a sets X, with composition.
(Σ + ,·), the collection of nonempty strings over Σ, with concatenation.
|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|
Size 1: 1
Size 2: 5
Size 3: 24
Size 4: 188
[Size 5]?: 1915
[Size 6]?: 28634
[Size 7]?: 1627672
[Semigroups in the Encyclopedia of Integer Sequences]