# Differences

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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> | ||

+ | )] |

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