## Lattice-ordered semigroups

Abbreviation: LSgrp

### Definition

A lattice-ordered semigroup (or $\ell$-semigroup) is a structure $\mathbf{A}=\langle A\vee,\wedge,\cdot\rangle$ of type $\langle 2,2,2\rangle$ such that

$\langle A,\vee,\wedge\rangle$ is a lattice

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

$\cdot$ distributes over $\vee$: $x(y\vee z)=xy\vee xz$, $(x\vee y)z=xz\vee yz$

Remark: This is a template. If you know something about this class, click on the Edit text of this page'' link at the bottom and fill out this page.

It is not unusual to give several (equivalent) definitions. Ideally, one of the definitions would give an irredundant axiomatization that does not refer to other classes.

##### Morphisms

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

### Definition

A is a structure $\mathbf{A}=\langle A,...\rangle$ of type $\langle ...\rangle$ such that

$...$ is …: $axiom$

$...$ is …: $axiom$

Example 1:

### Properties

Feel free to add or delete properties from this list. The list below may contain properties that are not relevant to the class that is being described.

Classtype variety yes yes

### Finite members

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

### Subclasses

[[Lattice-ordered monoids]] expanded type

### Superclasses

[[Semigroups]] reduced type
[[Lattices]] reduced type