Lower Separation Axioms for X-top Lattices
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| author | Abuhlail, J. Alfaraj, A. |
| author_facet | Abuhlail, J. Alfaraj, A. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_22355 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Lower Separation Axioms for X-top Lattices Abuhlail, J. Alfaraj, A. General Topology 2010: Primary 06A15, 6B30, Secondary: 13C13, 16D10, 16Y60 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. |
| title | Lower Separation Axioms for X-top Lattices |
| topic | General Topology 2010: Primary 06A15, 6B30, Secondary: 13C13, 16D10, 16Y60 |
| url | https://arxiv.org/abs/2510.22355 |