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Partially ordered semigroups

Abbreviation: PoSgrp

Definition

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

Morphisms

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: $h(x \cdot y)=h(x) \cdot h(y)$, $x\le y\Longrightarrow h(x)\le h(y)$

Examples

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

Properties

Classtype quasivariety

Finite members

$\begin{array}{lr} f(1)= &1\\ f(2)= &11\\ f(3)= &173\\ f(4)= &\\ f(5)= &\\ \end{array}$

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 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