semigroups - MathStructures

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=====Semigroups=====

Abbreviation: **Sgrp**
====Definition====
A \emph{semigroup} is a structure $\mathbf{S}=\langle S,\cdot
\rangle $, where $\cdot $ is an infix binary operation, called the 
\emph{semigroup product}, such that


$\cdot $ is associative:  $(xy)z=x(yz)$.
==Morphisms==
Let $\mathbf{S}$ and $\mathbf{T}$ be semigroups. A morphism from $\mathbf{S}$
to $\mathbf{T}$ is a function $h:S\to T$ that is a homomorphism: 

$h(xy)=h(x)h(y)$

====Examples====
Example 1: $\langle X^{X},\circ \rangle $, the collection of functions on a
sets $X$, with composition.

Example 1: $\langle \Sigma ^{+},\cdot \rangle $, the collection of nonempty
strings over $\Sigma $, with concatenation.



====Basic results====

====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 def. pr. cong.]]  |no |
^[[Amalgamation property]]  |no |
^[[Strong amalgamation property]]  |no |
^[[Epimorphisms are surjective]]  |no |
====Finite members====

$\begin{array}{lr}
f(1)= &1\\
f(2)= &5\\
f(3)= &24\\
f(4)= &188\\
f(5)= &1915\\
f(6)= &28634\\
f(7)= &1627672\\
f(8)= &3684030417\\
f(9)= &105\,978\,177\,936\,292\\
\end{array}$

[http://oeis.org/A027851 Semigroups in the Encyclopedia of Integer Sequences]

====Subclasses====
[[Bands]] 

[[Commutative semigroups]] 

[[Monoids]] 

[[Semigroups with zero]] 

====Superclasses====
[[Groupoids]] 

[[Partial semigroups]] 


====References====

[(Ln19xx>
)]

Trace: • finite_monoids • commutative_ordered_rings • distributive_lattice_ordered_semigroups • pseudocomplemented_distributive_lattices • ordered_monoids • action_algebras • continuous_lattices • idempotent_semirings_with_identity • commutative_residuated_partially_ordered_semigroups • order_algebras