Skip to main content
SpringerLink
Log in

Local completeness I

  • Published:
algebra universalis Aims and scope Submit manuscript

Abstract

An infinite algebra is locally complete if its local closure is the set of all finitary operations. We give a local completeness criterion in terms of a system R of finitary relations on A such that the polymorph Pol ρ of each ρ εR is locally incomplete and for every locally incomplete algebra 〈A; F〉 we have F ⊑ Pol ρ for some ρ ε R. This system consists of (i) certain natural relations whose polymorphs are best possible in the sense that they are co-atoms in the lattice of locally closed incomplete algebras, (ii) five types of binary relations, (iii) one type of ternary relations and (iv) at least ternary totally reflexive and symmetric relations that are not locally central.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. H. P. Gumm, Algebrasin permutable varieties: Geometrical properties of affine algebras, Algebra Universalis9 (1979) 8–34.

    Google Scholar 

  2. S. V. Jablonskii,Functional constructions in a k-valued logic (Russian), Trudy Mat. Inst. Steklov51 (1958) 5–142.

    Google Scholar 

  3. A. Knoebel,Further conclusions on functional completeness, Fundamenta Math.99 (1978) 93–112.

    Google Scholar 

  4. R. Pöschel,Concrete representation of algebraic structures and general Galois theory. In:Contribution to general algebra (Proc. Klagenfurt Conf. 1978), Verlag J. Heyn, Klagenfurt, Austria (1979) 249–972.

    Google Scholar 

  5. B. A. Romov,The Galois connection between the iterative Post algebras and relations on an infinite set, Kibernetika (Kiev)3 (1977) 62–64.

    Google Scholar 

  6. I. G. Rosenberg,Uber die funktionale Vollständigkeit in dem mehrwertigen Logiken (Struktur der Funktionen von mehreren Veränderlichen auf endlichen Mengen), Rozpravy Čs. Akademie Věd. Ser. Math. Nat. Sci.80 (1970) 3–93.

    Google Scholar 

  7. I. G. Rosenberg andD. Schweigert,Local clones, Elektron. Informationsverarb18 Kybernet.7/8 (1982) 389–401.

    Google Scholar 

  8. L. Szabó,Concrete representation of related structures of universal algebras I, Acta Sci. Math. (Szeged)40 (1978) 175–184.

    Google Scholar 

  9. A. Szendrei,On affine modules, In:Contributions to universal algebra, Colloq. Math. Soc. J. Bolyai, vol. 17, North Holland (1977), 457–464.

    Google Scholar 

  10. R. Quackenbush,A new proof of Rosenberg's primal algebra characterization theorem, In:Finite Algebra and Multiple-valued Logic (Proc. Conf. Szeged, 1979), Colloq. Math. Soc. J. Bolyai, vol. 28, North-Holland, Amsterdam, 1981; pp. 603–634.

    Google Scholar 

  11. I. G. Rosenberg,Uber die Verschiedenheit maximaler Klassen in P k , Rev. Roumaine Math. Pures Appl.14 (1969) 431–438.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Université de Montréal, Montréal, Quebec, Canada

    I. G. Rosenberg

  2. Attila József University, Szeged, Hungary

    L. Szabó

Authors
  1. I. G. Rosenberg
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. L. Szabó
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Rosenberg, I.G., Szabó, L. Local completeness I. Algebra Universalis 18, 308–326 (1984). https://doi.org/10.1007/BF01203368

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01203368

Keywords

Search

Navigation