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commutative_semigroups [2010/07/29 18:30] (current)
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 +=====Commutative semigroups=====
 +Abbreviation: **CSgrp**
 +====Definition====
 +A \emph{commutative semigroup} is a [[semigroups]] $\mathbf{S}=\langle
 +S,\cdot \rangle $ such that
 +
 +$\cdot $ is commutative:  $xy=yx$
 +====Definition====
 +A \emph{commutative 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:  $(xy)z=x(yz)$
 +
 +
 +$\cdot $ is commutative:  $xy=yx$
 +==Morphisms==
 +Let $\mathbf{S}$ and $\mathbf{T}$ be commutative semigroups. A morphism from
 +$\mathbf{S}$ to $\mathbf{T}$ is a function $h:Sarrow T$ that is a
 +homomorphism:
 +
 +$h(xy)=h(x)h(y)$
 +
 +====Examples====
 +Example 1: $\langle \mathbb{N},+\rangle $, the natural numbers, with additition.
 +
 +
 +
 +====Basic results====
 +
 +====Properties====
 +^[[Classtype]]  |variety |
 +^[[Equational theory]]  |decidable in polynomial time |
 +^[[Quasiequational theory]]  |decidable |
 +^[[First-order theory]]  | |
 +^[[Locally finite]]  |no |
 +^[[Residual size]]  | |
 +^[[Congruence distributive]]  |no |
 +^[[Congruence modular]]  |no |
 +^[[Congruence n-permutable]]  |no |
 +^[[Congruence regular]]  |no |
 +^[[Congruence uniform]]  |no |
 +^[[Congruence extension property]]  | |
 +^[[Definable principal congruences]]  | |
 +^[[Equationally def. pr. cong.]]  |no |
 +^[[Amalgamation property]]  |no |
 +^[[Strong amalgamation property]]  |no |
 +^[[Epimorphisms are surjective]]  |no |
 +====Finite members====
 +
 +$\begin{array}{lr}
 +[[Search for finite commutative semigroups]]
 +
 +f(1)= &1\\
 +f(2)= &3\\
 +f(3)= &12\\
 +f(4)= &58\\
 +f(5)= &325\\
 +f(6)= &2143\\
 +f(7)= &17291\\
 +\end{array}$
 +
 +====Subclasses====
 +[[Semilattices]]
 +
 +[[Commutative monoids]]
 +
 +====Superclasses====
 +[[Semigroups]]
 +
 +[[Partial commutative semigroups]]
 +
 +
 +====References====
 +
 +[(Ln19xx>
 +)]