Integral residuated lattices

Abbreviation: IRL

Definition

An integral residuated lattice is a residuated lattice $\mathbf{L}=\langle L, \vee, \wedge, \cdot, 1, \backslash, /\rangle$ that is

integral: $x\le 1$

Morphisms

Let $\mathbf{A}$ and $\mathbf{B}$ be integal residuated lattices. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism: $h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)$, $h(x\cdot y)=h(x)\cdot h(y)$, $h(x\backslash y)=h(x)\backslash h(y)$, $h(x/y)=h(x)/h(y)$, $h(1)=1$

Examples

Example 1: The negative cone of any l-group, e.g., $\mathbb Z^-$

Properties

Classtype variety decidable decidable undecidable no unbounded yes yes yes no yes no no no no

Finite members

$\begin{array}{lr} f(1)= &1\\ f(2)= &1\\ f(3)= &2\\ f(4)= &9\\ f(5)= &49\\ \end{array}$ $\begin{array}{lr} f(6)= &364\\ f(7)= &\\ f(8)= &\\ f(9)= &\\ f(10)= &\\ \end{array}$