In lattice theory, a bounded latticeL 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}.[1]
For instance, let Ln be a (flat) lattice with natomsa1, 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
^Insall, Matt (1992), "Some finiteness conditions in lattices—using nonstandard proof methods", Journal of the Australian Mathematical Society, Series A, 53 (2): 266–280, MR1175717
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