Closure operations, Continuous valuations on monoids and Spectral spaces

TL;DR

This paper introduces finite type closure operations and continuous valuations on monoids to construct and analyze spectral spaces, connecting algebraic and topological structures in monoid theory.

Contribution

It develops the theory of continuous valuations on monoids and demonstrates their role in forming spectral spaces, expanding the understanding of algebraic topology in monoid contexts.

Findings

Continuous valuations form spectral spaces on topological monoids.

Closure operations on monoids are classified and studied in detail.

Spectral spaces can be constructed from classes of monoids using these tools.

Abstract

We present several naturally occurring classes of spectral spaces using commutative algebra on pointed monoids. For this purpose, our main tools are finite type closure operations and continuous valuations on monoids which we introduce in this work. In the process, we make a detailed study of different closure operations on monoids. We prove that the collection of continuous valuations on a topological monoid with topology determined by any finitely generated ideal is a spectral space.