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## Distributive lattice-ordered semigroups

Abbreviation: **DLOS**

### Definition

A ** distributive lattice ordered 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 distributive lattice

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

$\cdot$ distributes over $\vee$: $x\cdot(y\vee z)=(x\cdot y)\vee (x\cdot z)$ and $(x\vee y)\cdot z=(x\cdot z)\vee (y\cdot z)$

##### Morphisms

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

### Examples

Example 1: Any collection $\mathbf A$ of binary relations on a set $X$ such that $\mathbf A$ is closed under union, intersection and composition.

Andreka 1991 AU proves that these examples generate the variety DLOS.

### Basic results

### 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.

### Finite members

$\begin{array}{lr} f(1)= &1\\ f(2)= &6\\ f(3)= &44\\ f(4)= &479\\ f(5)= &\\ \end{array}$ $\begin{array}{lr} f(6)= &\\ f(7)= &\\ f(8)= &\\ f(9)= &\\ f(10)= &\\ \end{array}$