Lower Separation Axioms for X-top Lattices

TL;DR

This work addresses how X-top lattices, which carry Zariski-like topologies on a chosen subset $X$, satisfy intermediate separation axioms between $T_{0}$ and $T_{1}$. It develops a coherent framework linking topological properties (Krull dimension, isolation, regularity) with lattice- and ring-theoretic structure, yielding precise criteria for when $X$ is $T_{\frac{1}{4}}$, $T_{\frac{1}{2}}$, or $T_{\frac{3}{4}}$, and relating these to notions like $ES$ and $CSI^{X}(X)$. The paper provides both sharp equivalences and constructive examples, including spectra of semirings such as $B(n,i)$, to illustrate when spectra exhibit particular quarter-separation properties or fail them. These results illuminate how spectral topologies interact with algebraic finiteness and dimensional invariants, with concrete implications for spectra of rings and semirings and for understanding the topology of prime, maximal, and minimal spectra in generalized settings.

Abstract

We study separation axioms for $X$-top-lattices (i.e. lattices $L$ 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 give graphical characterizations for an $X$-top-lattice to be $T_{1},$ $% T_{\frac{1}{4}},$ $T_{\frac{1}{2}},$ $T_{\frac{3}{4}}$ and provide several families of examples/counterexamples that illustrate our results. We apply our results mainly to the prime (resp. maximal, minimal) spectra of prime (resp. maximal, minimal) ideals of commutative (semi)rings.

Figures (2)

• Figure 1: The prime spectrum of$\mathbb{W}$
• Figure 2: Examples of trees and dual trees

Theorems & Definitions (83)

• Theorem 1.8
• Corollary 1.9
• Proof
• Definition 1.11
• Remark 1.17
• Definition 1.18
• Definition 2.3
• Lemma 2.4
• Proof
• Definition 2.5