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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>
		+	)]