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:
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.
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 |