- 0,1-simple lattice
In

lattice theory , abounded 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

*Wikimedia Foundation.
2010.*