Finite Lattices and Jordan-holder Sets 1)

In this paper we extend some aspects of the theory of 'supersolvable lattices' [3] to a more general class of finite lattices which includes the upper-semimodular lattices. In particular, all conjectures made in [33 concerning upper-semimodular lattices will be proved. For instance, we will prove that if L is finite upper-semimodular and if L' denotes L with any set of 'levels' removed, then the M6bius function of L' alternates in sign. Familiarity with [3] will be helpful but not essential for the understanding of the results of this paper. However, many of the proofs are identical to the proofs in [3-I (once the machinery has been suitably generalized) and will be omitted.