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—		cancellative_commutative_semigroups [2010/07/29 15:46] (current)
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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>
		+	)]
		+