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:
| Classtype | variety |
|---|---|
| Equational theory | |
| Quasiequational theory | undecidable |
| First-order theory | undecidable |
| Locally finite | no |
| Residual size | unbounded |
| Congruence distributive | yes |
| Congruence modular | yes |
| Congruence n-permutable | yes, n=2 |
| 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 |
$\begin{array}{lr}
f(1)= &1
f(2)= &1
f(3)= &3
f(4)= &20
f(5)= &115
f(6)= &899
f(7)= &7782
f(8)= &80468
\end{array}$