Abbreviation: IRL
An \emph{integral residuated lattice} is a residuated lattice $\mathbf{L}=\langle L, \vee, \wedge, \cdot, 1, \backslash, /\rangle$ that is
\emph{integral}: $x\le 1$
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: $h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)$, $h(x\cdot y)=h(x)\cdot h(y)$, $h(x\backslash y)=h(x)\backslash h(y)$, $h(x/y)=h(x)/h(y)$, $h(1)=1$
Example 1: The negative cone of any l-group, e.g., $\mathbb Z^-$
| Classtype | variety |
|---|---|
| Equational theory | decidable |
| Quasiequational theory | decidable |
| First-order theory | undecidable |
| Locally finite | no |
| Residual size | unbounded |
| Congruence distributive | yes |
| Congruence modular | yes |
| Congruence $n$-permutable | yes |
| Congruence regular | no |
| Congruence $e$-regular | yes |
| Congruence uniform | no |
| Congruence extension property | no |
| Definable principal congruences | no |
| Equationally def. pr. cong. | no |
| Amalgamation property | |
| Strong amalgamation property | |
| Epimorphisms are surjective |
$\begin{array}{lr}
f(1)= &1\\ f(2)= &1\\ f(3)= &2\\ f(4)= &9\\ f(5)= &49\\
\end{array}$ $\begin{array}{lr}
f(6)= &364\\ f(7)= &\\ f(8)= &\\ f(9)= &\\ f(10)= &\\
\end{array}$