Order analogues and Betti polynomials (Q1923997)

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.