Meet-semidistributive lattices

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Changed: 7c7
 the meet-semidistributive law SD∨:   x∨y = x∨z  ⇒  x∨y = x∨(y∧z).
 the meet-semidistributive law SD∧:   x∧y = x∧z  ⇒  x∧y = x∧(y∨z).

Definition

A meet-semidistributive lattice is a lattice L = (L,∨,∧) that satisfies the meet-semidistributive law SD:   xy = xz  ⇒  xy = x∧(yz).

Morphisms

Let L and M be meet-semidistributive lattices. A morphism from L to M is a function h : LM that is a homomorphism: h(xy) = h(x)∨h(y)  and  h(xy) = h(x)∧h(y).

Examples

D[d] = (D∪{d'},∨,∧), where D is any distributive lattice and d is an element in it that is split into two elements d,d' using Alan Day's doubling construction.

Properties

 Classtype quasivariety Equational theory Quasiequational theory First-order theory undecidable Congruence distributive yes Congruence modular yes Congruence n-permutable no Congruence regular no Congruence uniform no Congruence extension property Definable principal congruences Equationally definable principal congruences Amalgamation property no Strong amalgamation property no Epimorphisms are surjective Locally finite no Residual size unbounded

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Subclasses

Semidistributive lattices

Superclasses

Lattices

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