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., $\mathbb Z^-$

$\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}$

commutative integral residuated lattices

bounded integral residuated lattices

residuated lattices

integral lattice-ordered monoids