Skip to main content
SpringerLink
Log in

On normal and regular identities

  • Published:
algebra universalis Aims and scope Submit manuscript

Abstract

We deal with two kinds of special identities: normal and regular, considered by Mel'nik, Płonka and other authors. We point out fundamental properties of these identities. Also in §2 we show that the lattice of all subvarieties of the variety defined by all normal identities of a given varietyV (called the normal part of a varietyV) is isomorphic to the direct product of the lattice L(V) and a two-element chain. This result (Theorem 3) is a strengthening of a result of Mel'nik [14]. Theorem 4 states that the word problem for free algebras of the variety defined by all normal identities ofV is solvable if and only if it is solvable forV, which is due to the property of regular identities, proved in [8]. In §3 we consider normal and regular consequences of a given set of identities. Theorem 6 shows that for a given varietyV, satisfying a nonregular absorption law, the lattice L(Mod(NR(V))) is isomorphic to the direct product of the lattice L(V) and a four-element lattice, with two atoms.

Theorems in §4 collect some of results on the existence of a finite basis for normal and regular part of a given, finitely presented varietyV and of the finite basis property, as well, strengthening the result of Lakser, Padmanabhan and Platt [12].

Results above can be applied for semigroup varieties.

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. Birkhoff, G.,On the structure of abstract algebras. Proceedings of the Cambridge Phil. Soc.31 (1935), 434–454.

    Google Scholar 

  2. Dudek, J. andGraczyńska, E.,The lattice of varieties of algebras. Bull. Pol. Acad. Sci.29 (7, 8) (1981), 337–340.

    Google Scholar 

  3. Grätzer, G.,Universal Algebra. Springer-Verlag, Berlin, 1979.

    Google Scholar 

  4. Graczyńska, E.,On bases for normal identities. Studia Sci. Math. Hungarica19 (1984), 317–320.

    Google Scholar 

  5. Graczyńska, E.,On regular identities. Algebra Universalis17 (1983), 369–375.

    Google Scholar 

  6. Graczyńska, E.,On regular and symmetric identities I, II (abstracts). Bull, of the Section of Logic10 (3) (1981). 104–107;11 (3/4) (1983), 100–102.

    Google Scholar 

  7. Graczyńska, E.,Proofs of unary regular identities. Demonstraratio Math.16 (4) (1983), 925–929.

    Google Scholar 

  8. Graczyńska, E.,The word problem for regular identities. Beiträge zur Algebra and Geometrie24 (1987), 41–49.

    Google Scholar 

  9. Graczyńska, E. andPastijn, F.,Proofs of regular identities Houston J. of Math.8 (1) (1982), 61–67.

    Google Scholar 

  10. Graczyńska, E.,On normal and regular identities and hyperidentities, Proceedings of the V Universal Algebra Symposium, Turawa, Poland 3–7 May, 1988, World Sci. Publ. (1989), 107–135.

  11. Lakser, H., Padmanabhan, R. andPlatt, C. R.,Algebras defined by regular identities (abstract). Notices Amer. Math. Soc.17 (1970), 945.

    Google Scholar 

  12. Lakser, H., Padmanabhan, R. andPlatt, C. R.,Regular identities and finite basis property (abstract). Notices Amer. Math. Soc.21 (1974), A-528.

    Google Scholar 

  13. Mendelshon, N. S. andPadmanabhan, R.,A polynomial map preserving finite basis property. Journal of Algebra49 (1) (1977), 154–161.

    Google Scholar 

  14. Mel'nik, I. I.,Nilpotent shifts of varieties (in Russian). Mat. Zametki14 (5) (1973). MR 51 3028. English translation: Math. Notes14 (1973), 962–966.

  15. Mel'nik, I. I.,Normal closures of perfect varieties of universal algebras (in Russian).Ordered sets and lattices. Izdat. Saratov. Univ., Saratov (1971), 56–65. MR 52 3011.

    Google Scholar 

  16. Mel'nik, I. I.,On varieties of Ω-algebras (in Russian). Iss. po Algebre 1, Saratov (1969), 32–40. MR 40 7179.

  17. Płonka, J.,On equational classes of abstract algebras defined by regular equations. Fund. Math.64 (1969), 241–247.

    Google Scholar 

  18. Płonka, J.,On the sum of a system of disjoint unary algebras corresponding to a given type. Bull. Pol. Acad. Sci. (Math.)30 (7,8) (1982), 305–309.

    Google Scholar 

  19. Płonka, J.,On the lattice of unary algebras. Simon Stevin59 (1985), 353–364.

    Google Scholar 

  20. Płonka, J.,On strongly nonregular and trivializing varieties of algebras. Coll. Math. Soc. Bolyai43. Lectures in Univ. Algebra, Szeged (Hungary) (1983), 333–244.

    Google Scholar 

  21. Salii, W. N. Equationally normal varieties of semigroups (in Russian). Izvestija Wysszich Uczeb- nych Zavedenij, Matematika84 (1969), 61–68. MR 41 8555.

    Google Scholar 

  22. Salii, W. N.,A theorem of homomorphism of hard commutative bands of semigroups (in Russian). Teorija Polugrup i ee Prilozenija, Wypusk2 (1970), 69–74. MR 53 10959.

    Google Scholar 

  23. Tarski, A.,Equational logic and equational theories of algebras, in: H. A. Schmidt, K. Schütte and H. J. Thiele (eds.)Contributions to Mathematical Logic. North-Holland, Amsterdam, 1968, pp 275–288.

    Google Scholar 

  24. Taylor, W.,Equational logic. Houston J. of Math.5 (1979), 1–83.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, Polish Academy of Sciences, ul. Kopernika 18, 51-617, Wroclaw, Poland

    Ewa Graczyńska

Authors
  1. Ewa Graczyńska
    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

Graczyńska, E. On normal and regular identities. Algebra Universalis 27, 387–397 (1990). https://doi.org/10.1007/BF01190718

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01190718

Keywords

Search

Navigation