# Semigroups

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Difference (from revision 2 to current revision) (minor diff, author diff)

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#### Abbreviation: Sgrp

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 Variety

 variety

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 Decidable in polynomial time

 decidable in polynomial time

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 Undecidable

 undecidable

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 Undecidable

 undecidable

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 No

 no

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 Unbounded

 unbounded

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 No

 no

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 No

 no

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 No

 no

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 No

 no

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 No

 no

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 No

 no

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 No

 no

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 No

 no

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 No

 no

 Search for finite semigroups

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 [Size 7]?:  16276722
 [Size 7]?:  1627672 [Semigroups in the Encyclopedia of Integer Sequences]

 Semigroups with zero

### Definition

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).

### Morphisms

Let S and T be semigroups. A morphism from S to T is a function h : ST that is a homomorphism: h(xy) = h(x)h(y).

### Examples

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

(Σ + ,·), the collection of nonempty strings over Σ, with concatenation.

### 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

Search for finite semigroups

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]

### Subclasses

Bands
Commutative semigroups
Monoids
Semigroups with zero

### Superclasses

Groupoids
[Partial semigroups]?

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