{
  "id": "1407.0175",
  "version": "v2",
  "published": "2014-07-01T10:35:44.000Z",
  "updated": "2014-08-20T06:26:53.000Z",
  "title": "On structural completeness vs almost structural completeness problem: A discriminator varieties case study",
  "authors": [
    "Michal M. Stronkowski"
  ],
  "comment": "Added Proposition 5.4: Every minimal discriminator variety is minimal as a quasivariety. Added Example 5.11: Presents a minimal discriminator variety with a with a countably algebra which does not admit a homomorphism into any free algebra",
  "categories": [
    "math.LO"
  ],
  "abstract": "We study the following problem: Determine which almost structurally complete quasivarieties are structurally complete. We propose a general solution to this problem and then a solution in the semisimple case. As a consequence, we obtain a characterization of structurally complete discriminator varieties. An interesting corollary in logic follows: Let $L$ be a consistent propositional logic/deductive system in the language with formulas for verum, which is a theorem, and falsum, which is not a theorem. Assume also that $L$ has an adequate semantics given by a discriminator variety. Then $L$ is structurally complete if and only if it is maximal. All such logics/deductive systems are almost structurally complete.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-07-01T10:35:44.000Z",
      "abstract": "We study the following problem: Determine which almost structurally complete quasivarieties are structurally complete. We propose a general solution to this problem and then a solution in the semisimple case. In particular, we obtain a characterization of structurally complete discriminator varieties. They are known to be almost structural complete. A particularly interesting corollary in logic is: Let $L$ be a structurally complete propositional logic/deductive system in the language with formulas for verum, which is a theorem, and falsum, which is not a theorem. Assume also that $L$ has an adequate semantics given by a discriminator variety. Then $L$ must be minimal. All such logics/deductive systems are almost structurally complete. It shows the advantage of dealing with almost structural completeness instead of structural completeness.",
      "comment": "arXiv admin note: text overlap with arXiv:1402.5495",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-08-20T06:26:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "08C15",
      "03G25",
      "03B22",
      "08B20"
    ],
    "keywords": [
      "discriminator varieties case study",
      "discriminator variety",
      "structural completeness problem",
      "complete propositional logic/deductive system",
      "structurally complete discriminator varieties"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1407.0175C"
    }
  }
}