=====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====
^[[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]]  | |
====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====
[[Commutative distributive residuated lattices]] 

[[FLe-algebras]] 

====Superclasses====
[[Commutative multiplicative lattices]] 

[[Commutative residuated join-semilattices]] 

[[Commutative residuated meet-semilattices]] 

[[Residuated lattices]] 


====References====

[(Ln19xx>
)]