Higher Separation Axioms for $X$-top Lattices Applications to Commutative (Semi)rings
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866911371307778048 |
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| author | Abuhlail, Jawad Alfaraj, Abdulmushin |
| author_facet | Abuhlail, Jawad Alfaraj, Abdulmushin |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_08032 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Higher Separation Axioms for $X$-top Lattices Applications to Commutative (Semi)rings Abuhlail, Jawad Alfaraj, Abdulmushin Rings and Algebras Commutative Algebra General Topology 2020: Primary: 06B23, 6B30, 6B35, Secondary: 13C13, 16Y60 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. |
| title | Higher Separation Axioms for $X$-top Lattices Applications to Commutative (Semi)rings |
| topic | Rings and Algebras Commutative Algebra General Topology 2020: Primary: 06B23, 6B30, 6B35, Secondary: 13C13, 16Y60 |
| url | https://arxiv.org/abs/2601.08032 |