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Modular latticesModular lattices
A modular lattice is a lattice L = (L,∨,∧) that satisfies the modular identity: (( x∧z) ∨y) ∧z = ( x∧z) ∨( y∧z) .
A modular lattice is a lattice L = (L,∨,∧) that satisfies the modular law: x ≤ z ⇒ ( x∨y) ∧z ≤ x∨( y∧z) .
A modular lattice is a lattice L = (L,∨,∧) such that L has no sublattice isomorphic to the pentagon N5 =
Let L and M be modular 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).
M3 = is the smallest nondistributive modular lattice. By a result of Dedekind [1900] this lattice occurs as a sublattice of every nondistributive modular lattice.
| Classtype | Variety |
| Equational theory | Undecidable |
| Quasiequational theory | Undecidable |
| First-order theory | Undecidable |
| Congruence distributive | Yes |
| Congruence modular | Yes |
| Congruence n-permutable | No |
| Congruence regular | No |
| Congruence uniform | No |
| Congruence extension property | No |
| Definable principal congruences | No |
| Equationally definable principal congruences | No |
| Amalgamation property | No |
| Strong amalgamation property | No |
| Epimorphisms are surjective | No |
| Locally finite | No |
| Residual size | Unbounded |