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Distributive latticesDistributive lattices| yes, $\langle c,d\rangle\in Cg(a,b)\iff (a\wedge b)\wedge c=(a\wedge b)\wedge d \mbox{and} (a\vee b)\vee c=(a\vee b)\vee d |
| yes, (c,d) ∈ Cg(a,b) ⇔ (a∧b)∧c = (a∧b)∧d and (a∨b)∨c = (a∨b)∨d |
A distributive lattice is a lattice L = (L,∨,∧) such that
∧ distributes over ∨: x∧( y∨z) = ( x∧y) ∨( x∧z) .
A distributive lattice is a lattice L = (L,∨,∧) such that
∨ distributes over ∧: x∨( y∧z) = ( x∨y) ∧( x∨z) .
A distributive lattice is a lattice L = (L,∨,∧) such that
( x∧y) ∨( x∧z) ∨( y∧z) = ( x∨y) ∧( x∨z) ∧(
y∨z) .
A distributive lattice is a lattice L = (L,∨,∧) such that L has no sublattice isomorphic to the diamond M3 or the pentagon N5
Let L and M be distributive 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).
(P(S),∪,∩, ⊆ ), the collection of subsets of a sets S, ordered by inclusion.
| Classtype | variety |
| Equational theory | decidable |
| Quasiequational theory | decidable |
| First-order theory | undecidable |
| Congruence distributive | yes |
| Congruence modular | yes |
| Congruence n-permutable | no |
| Congruence regular | no |
| Congruence uniform | no |
| Congruence extension property | yes |
| Definable principal congruences | no |
| Equationally definable principal congruences | yes, (c,d) ∈ Cg(a,b) ⇔ (a∧b)∧c = (a∧b)∧d and (a∨b)∨c = (a∨b)∨d |
| Amalgamation property | yes |
| Strong amalgamation property | no |
| Epimorphisms are surjective | no |
| Locally finite | yes |
| Residual size | 2 |