[Home]Commutative residuated lattices

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Abbreviation: CRL

Definition

A commutative residuated lattice is a residuated lattice L = (L,∨,∧,·,e,\,/) such that

· is commutative:   xy = yx.

Remark:

Morphisms

Let L and M be commutative residuated lattices. A morphism from L to M is a function h : LM that is a homomorphism: h(xy) = h(x)∨h(y)  and  h(xy) = h(x)∧h(y)  and  h(x·y) = h(xh(y)  and  h(x\y) = h(x)\h(y)  and  h(x/y) = h(x)/h(y)   and  h(e) = e.

Some results

Examples

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 definable principal congruences No
Amalgamation property
Strong amalgamation property
Epimorphisms are surjective

Finite members

[Size 1]?:  1
[Size 2]?:  1
[Size 3]?:  3
[Size 4]?:  16
[Size 5]?:  100
[Size 6]?:  794

Subclasses

Commutative distributive residuated lattices
FLe-algebras

Superclasses

[Commutative multiplicative lattices]?
[Commutative residuated join-semilattices]?
[Commutative residuated meet-semilattices]?
Residuated lattices


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Last edited April 6, 2003 1:52 pm (diff)
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