A residuated lattice is a structure L = (L,∨,∧,·,e,\,/) of type (2,2,2,0,2,2) such that
(L,·,e) is a
(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.
Let L and M be residuated lattices. A morphism from L to M is a function h : L→M that is a homomorphism: h(x∨y) = h(x)∨h(y) and h(x∧y) = h(x)∧h(y) and h(x·y) = h(x)·h(y) and h(x\y) = h(x)\h(y) and h(x/y) = h(x)/h(y) and h(e) = e.
|Equational theory||Decidable [Hiroakira Ono, Yuichi Komori, Logics without the contraction rule, J. Symbolic Logic 50 (1985) 169--201 MRreview ZMATH] [implementation]|
|Congruence n-permutable||Yes, n=2|
|Congruence extension property||No|
|Definable principal congruences||No|
|Equationally definable principal congruences||No|
|Strong amalgamation property|
|Epimorphisms are surjective|