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

TL;DR

This paper investigates separation axioms in $X$-top lattices, providing conditions for various $T_i$ properties, and applies these findings to spectra of prime, maximal, and minimal ideals in commutative (semi)rings.

Contribution

It establishes necessary and sufficient conditions for $X$-top lattices to satisfy different separation axioms and applies these to algebraic structures like commutative (semi)rings.

Findings

Conditions for $X$-top lattices to be $T_2$, $T_3$, $T_{3.5}$, $T_4$, and $T_6$.

Examples and counterexamples illustrating the separation properties.

Applications to spectra of prime, maximal, and minimal ideals in commutative (semi)rings.

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.