0,1-simple lattice

0,1-simple lattice

In lattice theory, a bounded lattice "L" is called a 0,1-simple lattice if nonconstant lattice homomorphisms of "L" preserve the identity of its top and bottom elements. That is, if "L" is 0,1-simple and ƒ is a function from "L" to some other lattice that preserves joins and meets and does not map every element of "L" to a single element of the image, then it must be the case that ƒ-1(ƒ(0)) = {0} and ƒ-1(ƒ(1)) = {1}.

For instance, let "Ln" be a lattice with "n" atoms "a"1, "a"2, ..., "a""n", top and bottom elements 1 and 0, and no other elements. Then for "n" ≥ 3, "Ln" is 0,1-simple. However, for "n" = 2, the function ƒ that maps 0 and "a"1 to 0 and that maps "a"2 and 1 to 1 is a homomorphism, showing that "L"2 is not 0,1-simple.

External links

*mathworld|urlname = 01-SimpleLattice|title = 0,1-Simple Lattice|author=Matt Insall

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