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:

$\begin{array}{lr} f(1)= &1 f(2)= &1 f(3)= &3 f(4)= &16 f(5)= &100 f(6)= &794 f(7)= &7493 f(8)= &84961 \end{array}$

Commutative distributive residuated lattices

FLe-algebras

Commutative multiplicative lattices

Commutative residuated join-semilattices

Commutative residuated meet-semilattices

Residuated lattices