The M\"obius Function of Generalized Factor Order

TL;DR

This paper applies discrete Morse theory to compute the M"obius function of generalized factor order, extending previous results on ordinary factor order by incorporating partial orders on the alphabet.

Contribution

It introduces a generalized notion of factor order with partial alphabet orders and provides a recursive formula for the M"obius function using discrete Morse theory.

Findings

Derived a recursive formula for the M"obius function in generalized factor order.

Extended Bj"orner's results to cases with partial alphabet orders.

Utilized discrete Morse theory to gain new insights into factor order structures.

Abstract

We use discrete Morse theory to determine the M\"obius function of generalized factor order. Ordinary factor order on the Kleene closure A* of a set A is the partial order defined by letting u\leq w if w contains u as a subsequence of consecutive letters. The M\"obius function of ordinary factor order was determined by Bj\"orner. Using Babson and Hersh's application of Robin Forman's discrete Morse theory to lexicographically ordered chains, we are able to gain new understanding of Bj\"orner's result and its proof. We generalize the notion of factor order to take into account a partial order on the alphabet A and, relying heavily on discrete Morse theory, give a recursive formula in the case where each letter of the alphabet covers a unique letter.