Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2510.22355

Lower Separation Axioms for X-top Lattices

J. Abuhlail·A. Alfaraj
View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

math.GN

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

J. Abuhlail, A. Alfaraj. 2025-10-25. Lower Separation Axioms for X-top Lattices. https://arxiv.org/abs/2510.22355

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Maximal Center of Distances of Finite Ultrametric Spaces and Perfect Binary Trees

We investigate the finite ultrametric spaces $(X,d)$ that have a given cardinality of the center of distances and a minimal cardinality of the set $X$. It is shown that such spaces are isometric if and only if their centers of distances are the same. The representing trees of these spaces are characterized up to isomorphism.

math.GN↗
arXiv2026

A continuous $3$-distributive frame that is not $\omega$-distributive

We give a negative answer to the question, posed by Ern\'e, whether every $3$-distributive lattice is $\omega$-distributive. More precisely, we exhibit a continuous frame that is $\kappa$-distributive for every integer $\kappa\geq 2$, but is not a wide coframe. The frame is the open-set lattice of a compact, locally compact, countably based $T_0$ topological meet-semilattice, obtained from Lawson's construction in the logarithmic form described by Goubault-Larrecq. The failure of $\omega$-distributivity is witnessed by an explicit matrix with countably many nonempty finite rows: all row joins are the same nonzero element, whereas every choice of one entry from each row has meet zero. The same space answers negatively Ern\'e's accompanying question whether every $4$-web space is a wide web space. All properties of the construction needed for these conclusions are proved directly.

math.GN↗
arXiv2026

An overlooked weakening of perfect normality: Perfect regularity in spaces and locales

We introduce the notion of perfect regularity as an appropriate weakening of perfect normality, both for spaces and locales. Various characterizations are given, using Dedekind-MacNeille completions, injective hulls, and sublocales. We place the new class of perfectly regular frames among various well-studied classes of frames. We also introduce the construction of perfect regularization of a completely regular frame, compare it to Isbell's well-known booleanization construction, and argue that it is at least as important as the latter.

math.GN↗