This is an old revision of the document!

Partial semigroups

Abbreviation: PSgrp

Definition

A partial semigroup is a structure $\mathbf{A}=\langle A,\cdot\rangle$, where

$\cdot$ is a partial binary operation, i.e., $\cdot: A\times A\to A+\{*\})$ and

$\cdot$ is associative: $(x\cdot y)\cdot z\ne *$ or $x\cdot (y\cdot z)\ne *$ imply $(x\cdot y)\cdot z=x\cdot (y\cdot z)$.

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 $x\cdot y\ne *$ then $h(x \cdot y)=h(x) \cdot h(y)$

Examples

Example 1: The morphisms is a small category under composition.

Properties

Classtype first-order

Finite members

$\begin{array}{lr} f(1)= &2\\ f(2)= &12\\ f(3)= &90\\ f(4)= &960\\ f(5)= &\\ \end{array}$ $\begin{array}{lr} f(6)= &\\ f(7)= &\\ f(8)= &\\ f(9)= &\\ f(10)= &\\ \end{array}$