Skip to main content
SpringerLink
Log in

On isotone functions with the Substitution Property in distributive lattices

  • Published:
Order Aims and scope Submit manuscript

Abstract

A universal algebra isaffine complete if all functions satisfying the Substitution Property are polynomials (composed of the basic operations and the elements of the algebra). In 1962, the first author proved that a bounded distributive lattice is affine complete if and only if it does not contain a proper Boolean interval. Recently, M. Ploščica generalized this result to arbitrary distributive lattices.

In this paper, we introduce a class of functions on a latticeL, we call themID-polynomials, that derive from polynomials on the ideal lattice (resp., dual ideal lattice) ofL; they are isotone functions and satisfy the Substitution Property. We prove that for a distributive latticeL, all unary functions with the Substitution Property are ID-polynomials if and only ifL contains no proper Boolean interval.

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. Dorninger, D. and Eigenthaler, G. (1982) On compatible and order-preserving functions, inUniversal Algebra and Applications, Banach Center Publications, Vol. 9, Warsaw, pp. 97–104.

    Google Scholar 

  2. Grätzer, G. (1962) On Boolean functions (Notes on lattice theory. II),Rev. Math. Pures Appl. 7, 693–697.

    Google Scholar 

  3. Grätzer, G. (1978)General Lattice Theory, Academic Press, New York, NY; Birkhäuser Verlag, Basel, Akademie Verlag, Berlin.

    Google Scholar 

  4. Ploščica, M. (1994) Affine complete distributive lattices,Order 11, 385–390.

    Google Scholar 

  5. Schweigert, D. (1974) Über endliche, ordnungspolynomvollständige Verbände,Monatsh. Math. 78, 68–76.

    Google Scholar 

  6. Werner, H. (1971) Produkte von Kongruenzklassengeometrien universeller Algebren,Math. Z. 121, 111–140.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Manitoba, R3T 2N2, Winnipeg, Manitoba, Canada

    G. Grätzer

  2. Department of Mathematics, Transport Engineering Faculity, Technical University of Budapest, Müegyetem rkp. 9, 1111, Budapest, Hungary

    E. T. Schmidt

Authors
  1. G. Grätzer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. E. T. Schmidt
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by I. Rival

The research of the first author was supported by the NSERC of Canada. The research of the second author was supported by the Hungarian National Foundation for Scientific Research, under Grant No. 1903.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Grätzer, G., Schmidt, E.T. On isotone functions with the Substitution Property in distributive lattices. Order 12, 221–231 (1995). https://doi.org/10.1007/BF01111740

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01111740

Mathematics Subject Classifications (1991)

Key words

Search

Navigation