Abbreviation: PoSgrp

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

$\langle A,\cdot\rangle$ is a semigroup

$\langle G,\le\rangle$ is a partially ordered set

$\cdot$ is \emph{orderpreserving}: $x\le y\Longrightarrow xz\le yz \text{ and } zx\le zy$

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

Example 1: The natural numbers larger than 1, with addition, or with multiplication.

$\begin{array}{lr}

f(1)= &1\\
f(2)= &11\\
f(3)= &173\\
f(4)= &\\
f(5)= &\\

\end{array}$

Gajdos Kuril 2014 Ordered semigroups of size at most 7 and linearly ordered semigroups of size at most 10, Semigroup Forum

Number of elements 5 6 7, values below are for partially ordered semigroups.

Semigroups 198838 13457454 4207546916

Commutative semigroups 37248 1337698 71748346

Monoids 13371 504634 32113642

Bands 20305 494848 14349957

Regular semigroups 22419 546386 15842224

Inverse semigroups 2886 44275 830584

2-nilpotent semigroups 243 1533 12038

3-nilpotent semigroups 14150 2561653 3215028097

Commutative partially ordered semigroups

Lattice-ordered semigroups expanded type

Partially ordered groupoids