Lower Separation Axioms for X-top Lattices

TL;DR

This paper investigates various separation axioms within X-top lattices, a class of lattices with Zariski-like topologies, providing graphical characterizations and examples, especially in the context of spectra of ideals in commutative semirings.

Contribution

It introduces new graphical criteria for separation axioms in X-top lattices and applies these to spectra of ideals in commutative semirings, expanding understanding of their topological properties.

Findings

X-top lattices are generally T0 but not T2.

Graphical characterizations for T1, T1/4, T1/2, T3/4 separation axioms.

Examples and counterexamples illustrating the separation properties.

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.