{
  "annotation_id": "a0b0e5ed-1362-4cc9-a0bf-6336acb69d28",
  "date_created": "2026-02-17T05:53:15.040000Z",
  "date_modified": "2026-02-17T05:53:15.040000Z",
  "file_hash": "a551eddc4b22e562330acba41ef1380e88406a2273c18f784cf86d7bec7ddb20",
  "private": false,
  "record": {
    "abstract": "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.",
    "arxiv_id": "2601.08032",
    "authors": [
      "Jawad Abuhlail",
      "Abdulmushin Alfaraj"
    ],
    "categories": [
      "math.RA",
      "math.AC",
      "math.GN"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "title": "Higher Separation Axioms for $X$-top Lattices Applications to Commutative (Semi)rings",
    "url": "https://arxiv.org/abs/2601.08032",
    "version": "v1"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "7847f8f5-109b-4ab0-bd3c-f11a5270f61c",
    "id": "arXiv Dataset",
    "type": "Model",
    "variant": "snapshot-2026-01-17",
    "version": "0.1.0"
  },
  "user_id": 1000002
}