{
  "schema_version": 1,
  "problem_number": "OPG-57403",
  "title": "Two Problems of Porton on Reloids: Metamonovalued Reloids Are Monovalued, and S(S(f)) = S(f) Fails",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "A reloid from a set A to a set B is a filter on A × B. Reloids were introduced by V. Porton as a common generalisation of binary relations and uniformities. We settle two problems on reloids from Porton's book \"General Topology as Ordered Semigroup Actions\" which are also listed in the Open Problem Garden. First, every metamonovalued reloid is monovalued: if (g ⊓ h)∘f = (g∘f) ⊓ (h∘f) holds for the identity reloid g of B and the principal reloid h of the complement of the diagonal of B, then some member of the filter f is the graph of a partial function. Together with a theorem of Porton this shows that, for nonempty families, monovalued, metamonovalued and weakly metamonovalued reloids are the same. Second, for a reloid f from a set to itself let S(f) be the join of the powers f^n, n ≥ 0. We give such an f on a countable set with S(f)∘S(f) ≠ S(f) and S(S(f)) ≠ S(f). This answers Porton's question negatively and refutes two conjectures of the book. The identities do hold for principal reloids, in particular on finite sets, and for Porton's operator S*. A funcoid on the same set shows that the corresponding two conjectures for funcoids fail as well. This is an unrefereed note.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.GN",
    "math.CT"
  ],
  "keywords": [
    "reloid",
    "funcoid",
    "filter",
    "monovalued",
    "metamonovalued",
    "endomorphism series",
    "uniform space",
    "counterexample",
    "Open Problem Garden",
    "OPG-57403",
    "OPG-751",
    "math.GN",
    "math.CT",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-10-08",
  "publication_date": "2026-10-08",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-08",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/opg-57403/",
  "pdf_url": "https://eulersolve.org/papers/opg-57403/paper.pdf?v=4b00b0a80e18",
  "doi": "10.5281/zenodo.23245513",
  "zenodo_record_url": "https://zenodo.org/records/23245513",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Answers two problems of V. Porton that are listed in the Open Problem Garden (records OPG-57403 and OPG-751). Every metamonovalued reloid is monovalued (Conjecture 1363 of Porton's book in the copy of 16 August 2026), and the identities S(S(f)) = S(f) and S(f)∘S(f) = S(f) fail for a reloid on a countable set (Conjecture 1253, part 1°, and Conjecture 1254). A funcoid on the same set refutes the corresponding conjectures for funcoids, which are not records of the dataset. The results are elementary and belong to a theory developed essentially by its author in a self-published, regularly rebuilt book; the answers use its conventions (reverse inclusion order, improper filter allowed, S with the identity term), which the note states. Three independent AI-assisted verification runs checked the proofs line by line; no proof depends on a computation. Unrefereed; no priority claim is made.",
  "files": {
    "paper.pdf": {
      "sha256": "4b00b0a80e18a6191f375619222e7ac609c245c05fb7a42a96378f752bd65149"
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    "source.zip": {
      "sha256": "f50ae70ddc50c81233258b5b25018f8afc4adcf540eaf9d60f0399c95c484715"
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    "verification_report.md": {
      "sha256": "10fef83412f5fca4196e962bdb1f17b6330094001fa5db1970958d1a7ceaf624"
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  },
  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}