Residuated lattices

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Definition

A residuated lattice is a structure L = (L,∨,∧,·,e,\,/) of type (2,2,2,0,2,2) such that

(L,·,e) is a monoid,
(L,∨,∧) is a lattice,
\ is the left residual of ·:   y ≤ x\z  ⇔   xy ≤ z, and
/ is the right residual of ·:   x ≤ z/y  ⇔   xy ≤ z.

Remark:

Morphisms

Let L and M be 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 Decidable [Hiroakira Ono, Yuichi Komori, Logics without the contraction rule, J. Symbolic Logic 50 (1985) 169--201 MRreview ZMATH] [implementation] 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 No Definable principal congruences No Equationally definable principal congruences No Amalgamation property Strong amalgamation property Epimorphisms are surjective

Finite members

[Small residuated lattices]
[Size 1]?:  1
[Size 2]?:  1
[Size 3]?:  3
[Size 4]?:  20
[Size 5]?:  149
[Size 6]?:  1488
[Size 7]?:  18554
[Size 8]?:  295292

Subclasses

Commutative residuated lattices
FL-algebras

Superclasses

[Multiplicative lattices]?
[Residuated join-semilattices]?
[Residuated meet-semilattices]?

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