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+ | =====Cancellative commutative semigroups===== | ||

+ | |||

+ | Abbreviation: **CanCSgrp** | ||

+ | ====Definition==== | ||

+ | A \emph{cancellative commutative semigroup} is a [[commutative semigroup]] $\mathbf{S}=\langle | ||

+ | S,\cdot \rangle $ such that | ||

+ | |||

+ | $\cdot $ is \emph{cancellative}: $x\cdot z=y\cdot z\Longrightarrow x=y$ | ||

+ | ==Morphisms== | ||

+ | Let $\mathbf{S}$ and $\mathbf{T}$ be cancellative commutative semigroups. A morphism from | ||

+ | $\mathbf{S}$ to $\mathbf{T}$ is a function $h:S\rightarrow T$ 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]] |Quasivariety | | ||

+ | ^[[Equational theory]] | | | ||

+ | ^[[Quasiequational theory]] | | | ||

+ | ^[[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} | ||

+ | f(1)= &1\\ | ||

+ | f(2)= &\\ | ||

+ | f(3)= &\\ | ||

+ | f(4)= &\\ | ||

+ | f(5)= &\\ | ||

+ | f(6)= &\\ | ||

+ | f(7)= &\\ | ||

+ | \end{array}$ | ||

+ | |||

+ | ====Subclasses==== | ||

+ | [[Cancellative commutative monoids]] | ||

+ | |||

+ | ====Superclasses==== | ||

+ | [[Cancellative semigroups]] | ||

+ | |||

+ | [[Commutative semigroups]] | ||

+ | |||

+ | |||

+ | ====References==== | ||

+ | |||

+ | [(Ln19xx> | ||

+ | )] | ||

+ | |||

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