[Home]Complemented lattices

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Abbreviation: CdLat

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.

Some results

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

Finite members

[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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Last edited June 5, 2003 9:33 am (diff)
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