Abbreviation: DLOS

A \emph{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 $(x\vee y)\cdot z=(x\cdot z)\vee (y\cdot z)$

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: $h(x\vee y)=h(x) \vee h(y)$, $h(x\wedge y)=h(x) \wedge h(y)$, $h(x\cdot y)=h(x) \cdot h(y)$

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

H. Andreka¹⁾ proves that these examples generate the variety DLOS.

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.

$\begin{array}{lr}

f(1)= &1\\
f(2)= &6\\
f(3)= &44\\
f(4)= &479\\
f(5)= &\\

\end{array}$

Distributive lattice-ordered monoids

Commutative distributive lattice-ordered semigroups

Lattice-ordered semigroups