# Differences

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commutative_residuated_lattices [2010/07/29 15:46] (current)
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+=====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====
+
+$\begin{array}{lr} +f(1)= &1\\ +f(2)= &1\\ +f(3)= &3\\ +f(4)= &16\\ +f(5)= &100\\ +f(6)= &794\\ +\end{array}$
+
+====Subclasses====
+[[Commutative distributive residuated lattices]]
+
+[[FLe-algebras]]
+
+====Superclasses====
+[[Commutative multiplicative lattices]]
+
+[[Commutative residuated join-semilattices]]
+
+[[Commutative residuated meet-semilattices]]
+
+[[Residuated lattices]]
+
+
+====References====
+
+[(Ln19xx>
+)]

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