Abbreviation: CRL
A \emph{commutative residuated lattice} is a residuated lattice $\mathbf{L}=\langle L, \vee, \wedge, \cdot, e, \backslash, /\rangle $ such that
$\cdot$ is commutative: $xy=yx$
Remark:
Let $\mathbf{L}$ and $\mathbf{M}$ be commutative residuated lattices. A morphism from $\mathbf{L}$ to $\mathbf{M}$ is a function $h:L\rightarrow M$ that is a homomorphism:
$h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)$, $h(x\cdot y)=h(x)\cdot h(y)$, $h(x\backslash y)=h(x)\backslash h(y)$, $h(x/y)=h(x)/h(y)$, and $h(e)=e$
Example 1:
| Classtype | Variety |
|---|---|
| Equational theory | Decidable |
| 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 | Yes |
| Definable principal congruences | No |
| Equationally def. pr. cong. | No |
| Amalgamation property | |
| Strong amalgamation property | |
| Epimorphisms are surjective |
$\begin{array}{lr}
f(1)= &1
f(2)= &1
f(3)= &3
f(4)= &16
f(5)= &100
f(6)= &794
f(7)= &7493
f(8)= &84961
\end{array}$
Commutative multiplicative lattices
Commutative residuated join-semilattices