Let and be lattices. A map from to is called a lattice homomorphism if respects meet and join. That is, for ,
From this definition, one also defines lattice isomorphism, lattice endomorphism, lattice automorphism respectively, as a bijective lattice homomorphism, a lattice homomorphism into itself, and a lattice isomorphism onto itself.
If both and are bounded with lattice homomorphism , then is said to be a -lattice homomorphism if and are top and bottom of . In other words,
where are top and bottom elements of and respectively.
The idea behind these definitions comes from the idea of a homomorphism 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 and are both bounded lattices, a homomorphism between and must preserve and . Similarly, if only has and is bounded, then a homomorphism between them should preserve alone.
One can show that every Boolean algebra can be embedded into the power set of some set . That is, there is a one-to-one lattice homomorphism from into a Boolean subalgebra of (under the usual set union and set intersection operations) (see link below). If is in addition a complete lattice and an atomic lattice, then is lattice isomorphic to for some set .
|Date of creation||2013-03-22 15:41:31|
|Last modified on||2013-03-22 15:41:31|
|Last modified by||CWoo (3771)|