## Lattice-ordered monoids

Abbreviation: LMon

### Definition

A \emph{lattice-ordered monoid} (or \emph{$\ell$-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: $h(x \vee y)=h(x) \vee h(y)$, $h(x \wedge y)=h(x) \wedge h(y)$, $h(x \cdot y)=h(x) \cdot h(y)$, $h(1)=1$.

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

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

### Subclasses

[[Residuated lattices]] expanded type

### Superclasses

[[Lattice ordered semigroups]] reduced type