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## 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.

### Basic results

### Properties

### Finite members

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

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

### Subclasses

### Superclasses

### References

Trace: » partial_semigroups