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partial_semigroups [2016/11/26 17:18] (current)
jipsen created
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 +=====Partial semigroups=====
 +
 +Abbreviation: **PSgrp**
 +
 +====Definition====
 +A \emph{partial semigroup} is a structure $\mathbf{A}=\langle A,\cdot\rangle$, where
 +
 +$\cdot$ is a \emph{partial binary operation}: $\exists D\subseteq A\times A(\cdot:D\to A)$ and
 +
 +$\cdot$ is \emph{associative}: $(x\cdot y)\cdot z\in A$ implies $(x\cdot y)\cdot z=x\cdot (y\cdot z)$ and
 +
 +$x\cdot (y\cdot z)\in A$ implies $(x\cdot y)\cdot z=x\cdot (y\cdot z)$.
 +
 +Remark: $x\cdot y\in A\iff \langle x,y\rangle\in D$
 +
 +==Morphisms==
 +Let $\mathbf{A}$ and $\mathbf{B}$ be partial groupoids. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism:
 +if $x\cdot y\in A$ then $h(x \cdot y)=h(x) \cdot h(y)$
 +
 +====Examples====
 +Example 1:
 +
 +====Basic results====
 +
 +
 +====Properties====
 +
 +^[[Classtype]]                        |first-order  |
 +^[[Equational theory]]                | |
 +^[[Quasiequational theory]]           | |
 +^[[First-order theory]]               | |
 +^[[Locally finite]]                   | |
 +^[[Residual size]]                    | |
 +^[[Congruence distributive]]          | |
 +^[[Congruence modular]]               | |
 +^[[Congruence $n$-permutable]]        | |
 +^[[Congruence regular]]               | |
 +^[[Congruence uniform]]               | |
 +^[[Congruence extension property]]    | |
 +^[[Definable principal congruences]]  | |
 +^[[Equationally def. pr. cong.]]      | |
 +^[[Amalgamation property]]            | |
 +^[[Strong amalgamation property]]     | |
 +^[[Epimorphisms are surjective]]      | |
 +
 +====Finite members====
 +
 +$\begin{array}{lr}
 +  f(1)= &2\\
 +  f(2)= &12\\
 +  f(3)= &90\\
 +  f(4)= &960\\
 +  f(5)= &\\
 +\end{array}$    
 +$\begin{array}{lr}
 +  f(6)= &\\
 +  f(7)= &\\
 +  f(8)= &\\
 +  f(9)= &\\
 +  f(10)= &\\
 +\end{array}$
 +
 +
 +====Subclasses====
 +[[Semigroups]]
 +
 +[[Partial monoids]]
 +
 +
 +====Superclasses====
 +[[Partial groupoids]]
 +
 +
 +====References====
 +
 +