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Commutative residuated latticesCommutative residuated lattices
A commutative residuated lattice is a residuated lattice L = (L,∨,∧,·,e,\,/) such that
· is commutative: xy = yx.
Remark:
Let L and M be commutative 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 |
| 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 | Yes |
| Definable principal congruences | No |
| Equationally definable principal congruences | No |
| Amalgamation property | |
| Strong amalgamation property | |
| Epimorphisms are surjective |