Cancellative residuated lattices

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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.

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

None

Subclasses

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

Superclasses

Residuated lattices

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