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left_cancellative_semigroups [2010/07/29 15:46] (current)
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 +=====Left cancellative semigroups=====
 +
 +Abbreviation: **CanSgrp**
 +====Definition====
 +A \emph{left cancellative semigroup} is a semigroup $\mathbf{S}=\langle
 +S,\cdot \rangle $ such that
 +
 +$\cdot $ is left cancellative:  $z\cdot x=z\cdot y\Longrightarrow x=y$
 +==Morphisms==
 +Let $\mathbf{S}$ and $\mathbf{T}$ be left cancellative 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====
 +[[Left cancellative monoids]]
 +
 +[[Cancellative semigroups]]
 +
 +====Superclasses====
 +[[Semigroups]]
 +
 +
 +====References====
 +
 +[(Ln19xx>
 +)]