Partial semigroups

Abbreviation: PSgrp

Definition

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

$\cdot$ is a partial binary operation: $\exists D\subseteq A\times A(\cdot:D\to A)$ and

$\cdot$ is associative: $(x\cdot y)\cdot z\in A$ implies $(x\cdot y)\cdot z=x\cdot (y\cdot z)$ and

$x\cdot (y\cdot z)\in A$ implies $(x\cdot y)\cdot z=x\cdot (y\cdot z)$.

Remark: $x\cdot y\in A\iff \langle x,y\rangle\in D$

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 $x\cdot y\in A$ then $h(x \cdot y)=h(x) \cdot h(y)$

Example 1:

Properties

Classtype first-order

Finite members

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