Abbreviation: DRL

A \emph{distributive residuated lattice} is a residuated lattice $\mathbf{L}=\langle L, \vee, \wedge, \cdot, e, \backslash, /\rangle$ such that

$\vee, \wedge$ are distributive: $x\wedge(y\vee z) =(x\wedge y) \vee (x\wedge z)$

Remark:

Let $\mathbf{L}$ and $\mathbf{M}$ be distributive residuated lattices. A morphism from $\mathbf{L}$ to $\mathbf{M}$ is a function $h:L\rightarrow M$ that is a homomorphism:

$h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)$, $h(x\cdot y)=h(x)\cdot h(y)$, $h(x\backslash y)=h(x)\backslash h(y)$, $h(x/y)=h(x)/h(y)$, $h(e)=e$

Example 1:

$\begin{array}{lr} f(1)= &1 f(2)= &1 f(3)= &3 f(4)= &20 f(5)= &115 f(6)= &899 f(7)= &7782 f(8)= &80468 \end{array}$

Commutative distributive residuated lattices

Distributive FLe-algebras

Distributive multiplicative lattices

Residuated lattices