Abbreviation: PSgrp

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 $(x\cdot y)\cdot z=x\cdot (y\cdot z)$.

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 $h(x \cdot y)=h(x) \cdot h(y)$

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

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.

http://mathv.chapman.edu/~jipsen/uajs/PSgrp.html

$\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}$

Semigroups

Partial monoids

Partial groupoids