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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)) = and ƒ−1(ƒ(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.
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