Commutative residuated lattices

Abbreviation: CRL

Definition

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:

Morphisms

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 $h(e)=e$

Examples

Example 1:

Basic results

Properties

Finite members

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

Subclasses

Superclasses

References


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