Order analogues and Betti polynomials
type: Item
instance of: scholarly article
title: Order analogues and Betti polynomials
zbMATH Open document ID: 0860.20040
author: Lynne M. Butler
published in: Advances in Mathematics
publication date: 1997-04-16T00:00:00Z
review text: Given a graded rank \(n\) poset \(L\), its order theoretic \(q\)-analogues are families \(F=\{L(q)\}\) of finite graded rank \(n\) posets indexed by an infinite set of positive integers \(q\) with surjections \(\varphi:L(q)\to L\) such that: (i) \(H<K\) in \(L(q)\) implies \(\varphi(H)<\varphi(K)\); if \(\alpha\leq\varphi(K)\), then \(|\{H\mid H\leq K\) and \(\varphi(H)=\alpha\}|\) is a power of \(q\) determined by \(\alpha\) and \(\varphi(K)\); (iii) if \(\{\alpha\mid\alpha\leq\varphi(K)\}\) is a chain in \(L\), then \(\{H\mid H\leq K\}\) is a chain in \(L(q)\). This apparently quite complicated notion is applicable to many different situations already studied, where useful new combinatorial techniques may then be exploited. Thus, e.g., the lattice of subgroups of a finite abelian group is an order-theoretic analogue of the product of chains related to these groups. In analogy with the general setting, given such groups, an associated simplicial complex \(\Delta_s(p)\) whose simplices are chains of subgroups of order \(p^k\) for some \(k\in S=\{s_1,\dots,s_k\}\subseteq[n-1]\) can be studied from this viewpoint to determine \(\beta_s(p)\), the polynomial in \(p\) which enumerates the number of spheres of dimension \(k-1\) in the wedge of spheres to which \(\Delta_s(p)\) is homotopically equivalent. In particular, the strict surjections \(\varphi\) described above acquire topologically significant meaning, thereby providing a tight and natural explanation not only in the example of abelian groups used as a guide but in the case of order analogs of semimodular lattices which are themselves semimodular lattices also.
Mathematics Subject Classification ID: 20K01
Mathematics Subject Classification ID: 06A11
Mathematics Subject Classification ID: 20K27
Mathematics Subject Classification ID: 20D30
Mathematics Subject Classification ID: 05C10
Mathematics Subject Classification ID: 05B35
Mathematics Subject Classification ID: 06C10
Mathematics Subject Classification ID: 55U15
zbMATH DE Number: 934291
zbMATH Keywords: Betti polynomials
zbMATH Keywords: finite graded rank \(n\) posets
zbMATH Keywords: lattices of subgroups
zbMATH Keywords: finite Abelian groups
zbMATH Keywords: product of chains
zbMATH Keywords: simplicial complex
zbMATH Keywords: chains of subgroups
zbMATH Keywords: wedge of spheres
zbMATH Keywords: semimodular lattices
reviewed by: J. Neggers
MaRDI profile type: Publication
full work available at URL: aima.1996.0047
OpenAlex ID: W1977926338
DOI: 10.1006/AIMA.1996.0047
description: scientific article; zbMATH DE number 934291