The M\"obius function of generalized subword order

TL;DR

This paper provides a unified formula for the Mobius function of generalized subword order over any finite poset, extending previous results and analyzing the topological structure of intervals.

Contribution

It introduces a simple formula for the Mobius function of P* based on P's Mobius function, applicable to all finite posets, and characterizes the homotopy type of intervals for rank at most 1.

Findings

Unified formula for Mobius function of P*

Re-derivation of previous results in a uniform way

Determination of homotopy types for intervals in P* of rank ≤ 1

Abstract

Let P be a poset and let P* be the set of all finite length words over P. Generalized subword order is the partial order on P* obtained by letting u \leq w if and only if there is a subword u' of w having the same length as u such that each element of u is less than or equal to the corresponding element of u' in the partial order on P. Classical subword order arises when P is an antichain, while letting P be a chain gives an order on compositions. For any finite poset P, we give a simple formula for the Mobius function of P* in terms of the Mobius function of P. This permits us to rederive in a easy and uniform manner previous results of Bjorner, Sagan and Vatter, and Tomie. We are also able to determine the homotopy type of all intervals in P* for any finite P of rank at most 1.