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SemigroupsSemigroups
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.
| 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 |
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]