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.


  • 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.

Title lattice homomorphism
Canonical name LatticeHomomorphism
Date of creation 2013-03-22 15:41:31
Last modified on 2013-03-22 15:41:31
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 13
Author CWoo (3771)
Entry type Definition
Classification msc 06B05
Classification msc 06B99
Related topic OrderPreservingMap
Related topic RepresentingABooleanLatticeByFieldOfSets
Defines lattice isomorphism
Defines lattice endomorphism
Defines lattice automorphism
Defines 1}-latticehomomorphism