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}, i.e., $\cdot: A\times A\to A+\{*\}$ and

$\cdot$ is \emph{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.

Basic results

Partial semigroups can be identified with semigroups with zero since for any partial semigroup $A$ we can define a semigroup $A_0=A\cup\{0\}$ (assuming $0\notin A$) and extend the operation on $A$ to $A_0$ by $0x=0=x0$ for all $x\in A$. Conversely, given a semigroup with zero, say $B$, define a partial semigroup $A=B\setminus\{0\}$ and for $x,y\in A$ let $xy=*$ if $xy=0$ in $B$. These two maps are inverses of each other.

However, the category of partial semigroups is not the same as the category of semigroups with zero since the morphisms differ.

Properties

Classtype first-order

Finite members

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