Truncated Lattices

Abstract

How much does a (finite) lattice change when we remove the trivially always present elements 0 and 1? Pictorially there is very little change, since only the top and the bottom are gone. However in terms of order-theoretical properties there is a significant change. Note that both the proof of reconstructibility of finite lattices as well as the characterization of the fixed point property for lattices heavily relied on the existence of the smallest (or the largest) element. (We will again see the importance of the smallest element in Theorem 11.5.5, which settles the automorphism conjecture for finite lattices.) Thus in terms of two of our main open questions (and with respect to at least one future open question) 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.