The lexicographic sum of Cohen-Macaulay and shellable ordered sets
Since it is known that CM-posets are pure, i.e., all maximal chains have the same length, the posets under discussion here are taken to be finite and pure. If \({\mathcal P}=\{P_ q|\) \(q\in Q\}\) is a family of posets indexed by the elements of a poset Q, then the lexicographic sum of \({\mathcal P}\) over Q, denoted by \(\sum_{q\in Q}P_ q\), is the poset obtained by replacing each element \(q\in Q\) by the poset \(P_ q\). Called ''substitutions'' by the reviewer, lexicographic sums were studied in some detail by him [in Acta Math. Acad. Sci. Hung. 31, 233-258 (1978; Zbl 0395.05006)]. The following excellent result is the main theorem in this paper: T.4.1: The poset \(\sum_{q\in Q}P_ q\) is CM (over k) iff all the following conditions hold: (4.1) Each poset \(P_ q\) is CM; (4.2) The poset Q is CM; (4.3) If q and q' are distinct elements of Q having the same rank, then \(P_ q\) and \(P_{q'}\) are both antichains. The proof of the theorem is derived from a sequence of intermediate results, Lemmas 4.1-4.4, each interesting in their own right. The merit of the approach used is that one single definition of CM-ness, viz., Garsia's combinatorial/linear-algebraic characterization is used to prove this master theorem, thus providing a very general setting from which the student may depart in a study of the entire subject. The parallel concept(s) of (CL,EL,-) shellability are approached from a framework of thought similar to that used in the proofs of lemma 4.1-4.4 and of Theorem 4.1. Although not all dialogues for all shellability conditions work completely, CL-shellability, the weakest form of shellability-type designed so far, allows for an entire set of lemmas and a structure theorem precisely as in Theorem 4.1, demonstrating both the usefulness of the concept and its very strong relation to the entire CM point of view.
scientific article; zbMATH DE number 3912432