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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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