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cancellative_partial_monoids [2018/08/04 18:24] (current)
jipsen created
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 +=====Cancellative partial monoids=====
 +
 +Abbreviation: **CanPMon**
 +
 +====Definition====
 +A \emph{cancellative partial monoid} is a [[partial monoid]] such that
 +
 +$\cdot$ is \emph{left-cancellative}: $x\cdot y=x\cdot z\ne *$ implies $y=z$ and
 +
 +$\cdot$ is \emph{right-cancellative}: $x\cdot z=y\cdot z\ne *$ implies $x=y$.
 +
 +==Morphisms==
 +Let $\mathbf{A}$ and $\mathbf{B}$ be cancellative partial monoids. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism:
 +$h(e)=e$ and
 +if $x\cdot y\ne *$ 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====
 +
 +See http://mathv.chapman.edu/~jipsen/uajs/CanPMon.html
 +
 +$\begin{array}{lr}
 +  f(1)= &1\\
 +  f(2)= &2\\
 +  f(3)= &3\\
 +  f(4)= &9\\
 +  f(5)= &21\\
 +  f(6)= &125\\
 +  f(7)= &\\
 +  f(8)= &\\
 +  f(9)= &\\
 +  f(10)= &\\
 +\end{array}$
 +
 +====Subclasses====
 +[[Cancellative commutative partial monoids]]
 +
 +[[Cancellative monoids]]
 +
 +
 +====Superclasses====
 +[[Partial monoids]]
 +
 +
 +====References====
 +
 +