# Complemented lattices

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### Definition

A complemented lattice is a bounded lattice L = (L,∨,0,∧,1) such that every element has a complement:   y(xy = 1  and  xy = 0).

### Morphisms

Let L and M be complemented lattices. A morphism from L to M is a function h : LM that is a bounded lattice homomorphism: h(xy) = h(x)∨h(y)  and  h(xy) = h(x)∧h(y)  and  h(0) = 0  and  h(1) = 1.

### Examples

(P(S),∪,Ø,∩,S), the collection of subsets of a set S, with union, empty set, intersection, and the whole set S.

### Properties

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

[Size 1]?:  1
[Size 2]?:  1
[Size 3]?:  0
[Size 4]?:  1
[Size 5]?:  2
[Size 6]?:
[Size 7]?:
[Size 8]?:

### Subclasses

Complemented modular lattices

### Superclasses

Bounded lattices

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