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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>
 +)]