## 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

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