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 "L_n" 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, "L_n" 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