Abbreviation: OSgrp

An \emph{ordered semigroup} is a partially ordered semigroup $\mathbf{A}=\langle A,\cdot,\le\rangle$ such that

$\le$ is \emph{linear}: $x\le y\text{ or }y\le x$

Let $\mathbf{A}$ and $\mathbf{B}$ be ordered semigroups. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a orderpreserving homomorphism: $h(x \cdot y)=h(x) \cdot h(y)$, $x\le y\Longrightarrow h(x)\le h(y)$.

Example 1:

$\begin{array}{rr} f(1)=&1
f(2)=&6
f(3)=&44
f(4)=&386
f(5)=&3852
f(6)=&42640
f(7)=&516791
f(8)=&6817378
\end{array}$

http://oeis.org/A084965

Commutative ordered semigroups

Partially ordered semigroups

Chains reduced type