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.

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

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

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

References

1) Hajnal Andreka, Representations of distributive lattice-ordered semigroups with binary relations, Algebra Universalis 28 (1991), 12–25