Differences

This shows you the differences between two versions of the page.

clifford_semigroups [2010/07/29 15:46] (current)
Line 1: Line 1:
 +=====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>
 +)]