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

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 is said lowly finite if the set is finite for every element of . We mainly aim to provide a complete proof that, if is a subset of a complete lowly finite distributive lattice containing its join-irreducible elements, and an element of which is not join-irreducible, then belongs to the submodule of . 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.
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Taxonomy
TopicsAdvanced Algebra and Logic · semigroups and automata theory · Mathematical Dynamics and Fractals
