Higher Separation Axioms for $X$-top Lattices Applications to Commutative (Semi)rings

TL;DR

The paper develops higher separation axioms for $X$-top lattices, providing necessary and sufficient conditions for $X$ to satisfy $T_2$, $T_3$, $T_{3\frac12}$, $T_4$, $T_6$, and related notions, with a focus on spectra of commutative (semi)rings. It links topological properties (normality, complete normality, maximal retractability) to algebraic features (pm-property, max-retractability, Jacobson/dual Jacobson) and explores these in coatomic/atomic and spectral settings. Applications to $Spec(R)$, $Spec$ of primes, max/min spectra, and a suite of examples/counterexamples illustrate when regularity implies stronger separation axioms and when spectral spaces realize Stone-type structures in both rings and semirings. The work clarifies the interplay between algebraic spectra and classical topological separation, highlighting both positive characterizations and natural limitations via finite posets and semiring constructions.

Abstract

We study several separation axioms for $X$-top-lattices (i.e. a lattice $L$ for which a given subset $X\subseteq L\backslash \{1\}$ admits a \emph{% Zariski-like topology}). Such spaces are $T_{0}$ and usually far away from being $T_{2}.$ We provide sufficient/necessary conditions for an $X$-top lattice so that $X$ is $T_{2},$ \emph{regular} ($T_{3}$), \emph{completely regula}r ($T_{3\frac{1}{2}}$), \emph{normal}, \emph{completely normal} or \emph{perfectly normal} ($T_{6}$). We apply our results mainly to the spectrum of prime (resp. maximal, minimal) ideals of a commutative (semi)ring. We illustrate our results with several examples/counterexamples.

Figures (10)

• Figure 1: $N_5$: A non-distributive $X$-top lattice
• Figure 2: The prime spectrum of ${\mathbb{W}}$
• Figure 4: An $X$-lattice with $X$ extremely non-normal but not anti-normal
• Figure 5: An $X$-top lattice with $X$ completely normal but not perfectly normal
• Figure 6: A distributive modular lattice

Theorems & Definitions (61)

• Definition 1.4
• Theorem 1.6
• Corollary 1.7
• Example 1.9
• Definition 1.11
• Remark 1.14
• Lemma 1.15
• Definition 1.18
• Definition 1.19
• Definition 1.20