[Home]Cancellative residuated lattices

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Abbreviation: CanRL

Definition

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

· is right-cancellative:   xz = yz  ⇒  x = y, and
· is left-cancellative:   zx = zy  ⇒  x = y.

Remark:

Morphisms

Let L and M be cancellative 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
Quasiequational theory
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 No
Definable principal congruences No
Equationally definable principal congruences No
Amalgamation property
Strong amalgamation property
Epimorphisms are surjective

Finite members

None

Subclasses

[Cancellative commutative residuated lattices]?
[Cancellative distributive residuated lattices]?

Superclasses

Residuated lattices


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Last edited February 26, 2003 11:44 pm (diff)
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