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}. Insall Matt 2 Journal of the Australian Mathematical Society, Series A 1175717 266–280 Some finiteness conditions in lattices—using nonstandard proof methods 53 1992 10.1017/S1446788700035849 For instance, let Ln be a (flat) lattice with n atoms a1, a2, ..., an, 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 a1 to 0 and that maps a2 and 1 to 1 is a homomorphism, showing that L2 is not 0,1-simple. References Category:Lattice theory