Lattice-ordered monoids

Abbreviation: LMon

Definition

A lattice-ordered monoid (or $\ell$-monoid) is a structure $\mathbf{A}=\langle A\vee,\wedge,\cdot,1\rangle$ of type $\langle 2,2,2,0\rangle$ such that

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

$\langle A,\cdot,1\rangle$ is a monoid

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

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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 monoids. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism: $h(x \vee y)=h(x) \vee h(y)$, $h(x \wedge y)=h(x) \wedge h(y)$, $h(x \cdot y)=h(x) \cdot h(y)$, $h(1)=1$.

Examples

Example 1:

Basic results

Properties

Finite members

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

Subclasses

[[Residuated lattices]] expanded type

Superclasses

[[Lattice ordered semigroups]] reduced type

References