How much does a (finite) lattice change when we remove the (trivially always present) elements 0 and 1? With only the top and bottom elements gone, the picture does not change much at all. However, in terms of order-theoretical properties there is a significant change. Note that both the proof of reconstructibility of finite lattices (Corollary 8.6) as well as the characterization of the fixed point property for lattices (Theorem 8.10) heavily relied on the existence of the smallest (or the largest) element. The smallest element was important in Theorem 8.34, which settles the automorphism conjecture for finite lattices, too (see Exercise 9-8). Thus, in terms of three of our main open questions, the loss of 0 and 1 is significant. The question arises what “intrinsic” parts of the lattice structure can be used to tackle problems such as reconstruction or the fixed point property. To this end, in this chapter we investigate lattices from which top and bottom element have been removed.
KeywordsSimplicial Complex Algebraic Topology Comparability Graph Point Property Finite Lattice
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