A Survey of the Valuation Algebra motivated by a Fundamental Application to Dissection Theory

TL;DR

This paper surveys valuation algebra with a focus on its application to dissection theory, providing a complete proof of a key property in lowly finite distributive lattices and demonstrating its use in face counting for submanifold arrangements.

Contribution

It offers a complete proof of a property in lowly finite distributive lattices crucial for dissection theory, extending Zaslavsky's finite case to a broader setting.

Findings

Proved a key property for lowly finite distributive lattices

Extended Zaslavsky's finite lattice result to infinite cases

Applied the theorem to face counting in submanifold arrangements

Abstract

A lattice $L$ is said lowly finite if the set $[\mathsf{0},a]$ is finite for every element $a$ of $L$. We mainly aim to provide a complete proof that, if $M$ is a subset of a complete lowly finite distributive lattice $L$ containing its join-irreducible elements, and $a$ an element of $M$ which is not join-irreducible, then $\displaystyle \sum_{b \in M \cap [\mathsf{0},a]} \mu_M(b,a)b$ belongs to the submodule $\langle a \wedge b + a \vee b - a - b\ |\ a,b \in L \rangle$ of $\mathbb{Z}L$. That property was originally established by Zaslavsky for finite distributive lattice. It is essential to prove the fundamental theorem of dissection theory as will be seen. We finish with a concrete application of that theorem to face counting for submanifold arrangements.