Stern, Manfred. Semimodular lattices: theory and applications. / Manfred Stern. p. cm. – (Encyclopedia of mathematics and its applications ; v. 73) Includes bibliographical references and index.

Let P be a poset. A subset A of P is a k-family ii A contains no (k+1)-element chain. For i 1, let A i be the set of elements of A at depth i?1 in A. The k-families of P can be ordered by deening A B ii for all i, A i is included in the order ideal generated by B i. This paper examines minimal r-element k-families, deened as k-families A such that A = r and for every B < A, B < r. Minimal k-fam...

In this paper we study shellable posets (partially ordered sets), that is, finite posets such that the simplicial complex of chains is shellable. It is shown that all admissible lattices (including all finite semimodular and supersolvable lattices) and all bounded locally semimodular finite posets are shellable. A technique for labeling the edges of the Hasse diagram of certain lattices, due to...

A recent result of G. Czédli and E. T. Schmidt gives a construction of slim (planar) semimodular lattices from planar distributive lattices by adding elements, adding “forks”. We give a construction that accomplishes the same by deleting elements, by “resections”.

The notion of CD-independence is introduced as follows. A subset X of a lattice L with 0 is called CD-independent if for any x, y ∈ X , either x ≤ y or y ≤ x or x ∧ y = 0. In other words, if any two elements of X are either Comparable or Disjoint. Maximal CD-independent subsets are called CD-bases. The main result says that any two CD-bases of a finite distributive lattice L have the same numbe...