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单词 LatticeHomomorphism
释义

lattice homomorphism


Let L and M be lattices. A map ϕ from L to M is called a lattice homomorphismMathworldPlanetmath if ϕ respects meet and join. That is, for a,bL,

  • ϕ(ab)=ϕ(a)ϕ(b), and

  • ϕ(ab)=ϕ(a)ϕ(b).

From this definition, one also defines lattice isomorphism, lattice endomorphism, lattice automorphism respectively, as a bijectiveMathworldPlanetmathPlanetmath lattice homomorphism, a lattice homomorphism into itself, and a lattice isomorphism onto itself.

If in addition L is a bounded latticeMathworldPlanetmath with top 1 and bottom 0, with ϕ and M defined as above, then ϕ(a)=ϕ(1a)=ϕ(1)ϕ(a), and ϕ(a)=ϕ(0a)=ϕ(0)ϕ(a) for all aL. Thus L is mapped onto a boundedPlanetmathPlanetmathPlanetmathPlanetmath sublattice ϕ(L) of M, with top ϕ(1) and bottom ϕ(0).

If both L and M are bounded with lattice homomorphism ϕ:LM, then ϕ is said to be a {0,1}-lattice homomorphism if ϕ(1) and ϕ(0) are top and bottom of M. In other words,

ϕ(1L)=1M   and   ϕ(0L)=0M,

where 1L,1M,0L,0M are top and bottom elements of L and M respectively.

Remarks.

  • The idea behind these definitions comes from the idea of a homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath between two algebraic systems of the same type. We require the the homomorphism to preserve all finitary operations, including the nullary ones. This means that if the algebraic system contains constants, they need to be preserved under the homomorphism. Thus, if L and M are both bounded lattices, a homomorphism between L and M must preserve 0 and 1. Similarly, if L only has 0 and M is bounded, then a homomorphism between them should preserve 0 alone.

  • In the case of completePlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath lattices, there are operationsMathworldPlanetmath that are infinitary, so the homomorphism between two complete lattices should preserve the infinitary operations as well. The resulting lattice homomorphism is a complete lattice homomorphism.

  • One can show that every Boolean algebraMathworldPlanetmath B can be embedded into the power setMathworldPlanetmath of some set S. That is, there is a one-to-one lattice homomorphism ϕ from B into a Boolean subalgebra of 2S (under the usual set union and set intersectionMathworldPlanetmathPlanetmath operations) (see link below). If B is in addition a complete latticeMathworldPlanetmath and an atomic lattice, then B is lattice isomorphic to 2S for some set S.

Titlelattice homomorphism
Canonical nameLatticeHomomorphism
Date of creation2013-03-22 15:41:31
Last modified on2013-03-22 15:41:31
OwnerCWoo (3771)
Last modified byCWoo (3771)
Numerical id13
AuthorCWoo (3771)
Entry typeDefinition
Classificationmsc 06B05
Classificationmsc 06B99
Related topicOrderPreservingMap
Related topicRepresentingABooleanLatticeByFieldOfSets
Defineslattice isomorphism
Defineslattice endomorphism
Defineslattice automorphism
Defines
Defines1}-latticehomomorphism
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更新时间:2025/5/4 2:50:52