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—		clifford_semigroups [2010/07/29 15:46] (current)
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		+	=====Clifford semigroups=====
		+	Abbreviation: **CliffSgrp**
		+	====Definition====
		+	A \emph{Clifford semigroup} is an [[inverse semigroups]] $\mathbf{S}=\langle
		+	S,\cdot,^{-1}\rangle $ that is also [[completely regular semigroups]].
		+	====Definition====
		+	A \emph{Clifford semigroup} is a structure $\mathbf{S}=\langle
		+	S,\cdot,^{-1}\rangle $ such that
		+
		+
		+	$\cdot$ is associative:  $(xy)z=x(yz)$
		+
		+
		+	$^{-1}$ is an inverse:  $xx^{-1}x=x$, $(x^{-1})^{-1}=x$
		+
		+
		+	$xx^{-1}=x^{-1}x$, $xx^{-1}y^{-1}y=y^{-1}yxx^{-1}$, $xx^{-1}=x^{-1}x$
		+	==Morphisms==
		+	Let $\mathbf{S}$ and $\mathbf{T}$ be Clifford semigroups. A morphism from
		+	$\mathbf{S}$ to $\mathbf{T}$ is a function $h:S\rightarrow T$ that is a
		+	homomorphism:
		+
		+	$h(xy)=h(x)h(y)$, $h(x^{-1})=h(x)^{-1}$
		+
		+	====Examples====
		+	Example 1:
		+
		+	====Basic results====
		+
		+	====Properties====
		+	^[[Classtype]]  |Variety |
		+	^[[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]]  |No |
		+	^[[Definable principal congruences]]  | |
		+	^[[Equationally def. pr. cong.]]  |No |
		+	^[[Amalgamation property]]  |No |
		+	^[[Strong amalgamation property]]  |No |
		+	^[[Epimorphisms are surjective]]  |Yes |
		+	====Finite members====
		+
		+	$\begin{array}{lr}
		+	f(1)= &1\\
		+	f(2)= &\\
		+	f(3)= &\\
		+	f(4)= &\\
		+	f(5)= &\\
		+	f(6)= &\\
		+	f(7)= &\\
		+	\end{array}$
		+
		+	====Subclasses====
		+	[[Groups]]
		+
		+	====Superclasses====
		+	[[Completely regular semigroups]]
		+
		+	[[Inverse semigroups]]
		+
		+
		+	====References====
		+
		+	[(Ln19xx>
		+	)]