# Suprema in spectral spaces and the constructible closure

## TL;DR

This paper explores the relationship between suprema in spectral spaces, the constructible topology, and applications to algebraic lattices, rings, and domains, providing new topological characterizations and density results.

## Contribution

It introduces conditions under which suprema exist within the constructible closure in spectral spaces and applies these findings to algebraic lattices, rings, and domains.

## Key findings

- Conditions for the existence of suprema in constructible closures
- Density properties of spaces of rings and ideals
- Topological characterizations of certain domains

## Abstract

Given an arbitrary spectral space $X$, we endow it with its specialization order $\leq$ and we study the interplay between suprema of subsets of $(X,\leq)$ and the constructible topology. More precisely, we investigate about when the supremum of a set $Y\subseteq X$ exists and belongs to the constructible closure of $Y$. We apply such results to algebraic lattices of sets and to closure operations on them, proving density properties of some distinguished spaces of rings and ideals. Furthermore, we provide topological characterizations of some class of domains in terms of topological properties of their ideals.