{
  "id": "2101.03647",
  "version": "v1",
  "published": "2021-01-11T00:10:50.000Z",
  "updated": "2021-01-11T00:10:50.000Z",
  "title": "Absolutely Free Hyperalgebras",
  "authors": [
    "Marcelo E. Coniglio",
    "Guilherme V. Toledo"
  ],
  "comment": "16 pages",
  "categories": [
    "math.LO",
    "math.CT"
  ],
  "abstract": "It is well know from universal algebra that, for every signature $\\Sigma$, there exist algebras over $\\Sigma$ which are freely generated. Furthermore, they are, up to isomorphisms, unique, and equal to algebras of terms. Equivalently, the forgetful functor, from the category of $\\Sigma$-algebras to $\\textbf{Set}$, has a left adjoint. This result does not extend to hyperalgebras, which generalize algebras by allowing the result of an operation to assume a non-empty set of values. Not only freely generated hyperalgebras do not exist, but the forgetful functor $\\mathcal{U}$, from the category of $\\Sigma$-hyperalgebras to $\\textbf{Set}$, does not have a left adjoint. In this paper we generalize, in a natural way, algebras of terms to hyperalgebras of terms, which display many properties of freely generated algebras: they extend uniquely to homomorphisms, not functions, but pairs of functions and collections of choices, which select how an homomorphism approaches indeterminacies; and they are generated by a set that fits a strong definition of basis, which we call the ground of the hyperalgebra. With these definitions at hand, we offer simplified proofs that freely generated hyperalgebras do not exist and that $\\mathcal{U}$ does not have a left adjoint.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-01-11T00:10:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03B99",
      "08C99",
      "08B20"
    ],
    "keywords": [
      "absolutely free hyperalgebras",
      "left adjoint",
      "freely generated hyperalgebras",
      "forgetful functor",
      "homomorphism approaches indeterminacies"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}