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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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