# Differences

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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>
+)]