# Differences

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commutative_inverse_semigroups [2010/07/29 15:46] (current)
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+=====Commutative inverse semigroups=====
+Abbreviation: **CInvSgrp**
+====Definition====
+A \emph{commutative inverse semigroup} is an [[inverse semigroups]] $\mathbf{S}=\langle +S,\cdot,^{-1}\rangle$ such that
+
+$\cdot$ is commutative:  $xy=yx$
+====Definition====
+A \emph{commutative inverse semigroup} is a structure $\mathbf{S}=\langle +S,\cdot,^{-1}\rangle$ such that
+
+
+$\cdot$ is associative:  $(xy)z=x(yz)$
+
+
+$\cdot$ is commutative:  $xy=yx$
+
+
+$^{-1}$ is an inverse:  $xx^{-1}x=x$, $(x^{-1})^{-1}=x$
+==Morphisms==
+Let $\mathbf{S}$ and $\mathbf{T}$ be commutative inverse 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]]  |Yes |
+^[[Definable principal congruences]]  | |
+^[[Equationally def. pr. cong.]]  |No |
+^[[Amalgamation property]]  |Yes |
+^[[Strong amalgamation property]]  |Yes |
+^[[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====
+[[Abelian groups]]
+
+[[Semilattices]]
+
+====Superclasses====
+[[Inverse semigroups]]
+
+
+====References====
+
+[(Ln19xx>
+)]

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