Spectrality and monoids

TL;DR

This work addresses the spectrality of spaces constructed from monoids by endowing the set of proper ideals $\mathcal{I}^+_M$ with the coarse lower topology. It provides a self-contained proof that $\mathcal{I}^+_M$ is a spectral space by proving quasi-compactness via the Alexander Subbasis Theorem, showing sobriety, and verifying a suitable basis of compact open sets, while leveraging the algebraic lattice structure of $\mathcal{I}_M$ and the openness of $\mathcal{I}^+_M$ in $\mathcal{I}_M$. The approach avoids heavy reliance on basis-existence arguments through a targeted lemma, and highlights the role of the monoid identity in ensuring quasi-compactness. Overall, the paper extends spectral-space concepts to the setting of monoids with a topology-driven perspective on ideals, enriching the interplay between lattice theory and topological structure.

Abstract

We prove that the set of proper ideals of a monoid endowed with coarse lower topology is a spectral space.

Theorems & Definitions (3)

• Theorem 3.1
• Lemma 3.2
• proof