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Studia Logica

, Volume 54, Issue 3, pp 371–389 | Cite as

Joins of minimal quasivarieties

  • M. E. Adams
  • W. Dziobiak
Article

Abstract

LetL(K) denote the lattice (ordered by inclusion) of quasivarieties contained in a quasivarietyK and letD2 denote the variety of distributive (0, 1)-lattices with 2 additional nullary operations. In the present paperL(D2) is described. As a consequence, ifM+N stands for the lattice join of the quasivarietiesM andN, then minimal quasivarietiesV0,V1, andV2 are given each of which is generated by a 2-element algebra and such that the latticeL(V0+V1), though infinite, still admits an easy and nice description (see Figure 2) while the latticeL(V0+V1+V2), because of its intricate inner structure, does not. In particular, it is shown thatL(V0+V1+V2) contains as a sublattice the ideal lattice of a free lattice with ω free generators. Each of the quasivarietiesV0,V1, andV2 is generated by a 2-element algebra inD2.

Key words

Quasivariety lattice of quasivarieties distributive (0, 1)-lattice nullary operations free lattice Priestley space graph 

1991 Mathematics Subject Classification

Primary: 06D99, 08C15 Secondary: 06B25 

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References

  1. Adams, M. E. andW. Dziobiak, 1994, ‘Q-universal quasivarieties of algebras’,Proc. Amer. Math. Soc. 120, 1053–1059.Google Scholar
  2. Adams, M. E., W. Dziobiak, M. Gould, andJ. Schmid, [a], ‘Quasivarieties of pseudocomplemented semilattices’Fund. Math., to appear.Google Scholar
  3. Adams, M. E., V. Koubek, andJ. Sichler, 1985, ‘Homomorphisms and endomorphisms of distributive lattices’Houston J. Math. 11, 129–145.Google Scholar
  4. Birkhoff, G., 1946 (1945), ‘Universal algebra’ in:Proc. First. Canadian Math. Congress, Montréal 1945 University of Toronto Press, Toronto, 310–326.Google Scholar
  5. Burris, S. andH.P. Sankappanavar, 1981,A Course in Universal Algebra Springer-Verlag, New York.Google Scholar
  6. Gorbunov, V. A., [a], ‘The structure of lattices of quasivarieties’Algebra Universalis, to appear.Google Scholar
  7. Gorbunov, V. A. andV.I. Tumanov, 1980, ‘A class of lattices of quasivarieties’ (in Russian),Algebra i Logika 19, 59–80.Google Scholar
  8. Grätzer, G., H. Lakser, andR.W. Quackenbush, 1980, ‘On the lattice of quasivarieties of distributive lattices with pseudocomplementation’Acta Sci. Math. (Szeged)42, 257–263.Google Scholar
  9. Hedrlín, Z. andA. Pultr, 1966, ‘On full embeddings of categories of algebras’Illinois J. Math. 10, 392–406.Google Scholar
  10. Kartashov, V. K., 1985, ‘On lattices of quasivarieties of unars’ (in Russian),Sibirsk. Mat. Zh. 26, 49–62.Google Scholar
  11. Koubek, V., 1985, ‘Infinite image homomorphisms of distributive bounded lattices’Colloq. Math. Soc. János Bolyai 43, Lectures in Universal Algebra, Szeged (Hungary) 1983 North-Holland, Amsterdam, 241–281.Google Scholar
  12. Mal'cev, A. I., 1968, ‘On certain frontier questions in algebra and mathematical logic’, (in Russian),Proc. Int. Congr. Mathematicians, Moscow 1966, Mir, 217–231.Google Scholar
  13. Mal'cev, A. I., 1973,Algebraic Systems Grundlehren der Mathematischen Wissenschaften 192, Springer-Verlag, New York.Google Scholar
  14. Priestley, H. A., 1970, ‘Representation of distributive lattices by means of ordered Stone spaces’Bull. London Math. Soc. 2, 186–190.Google Scholar
  15. Pultr, A. andV. Trnková, 1980,Combinatorial, Algebraic and Topological Representations of Groups, Semigroups and Categories North-Holland, Amsterdam.Google Scholar
  16. Sapir, M. V., 1984, ‘Varieties with a countable number of subquasivarieties’ (in Russian),Sibirsk. Mat. Zh. 25, 148–163.Google Scholar
  17. Sapir, M. V., 1985, ‘The lattice of quasivarieties of semigroups’Algebra Universalis 21, 172–180.Google Scholar
  18. Shafaat, A., 1974, ‘On implicational completeness’Canad. J. Math. 26, 761–768.Google Scholar
  19. Tropin, M. P., 1983, ‘An embedding of a free lattice into the lattice of quasivarieties of distributive lattices with pseudocomplementation’ (in Russian),Algebra i Logika 22, 159–167.Google Scholar

Copyright information

© Kluwer Academic Publishers 1995

Authors and Affiliations

  • M. E. Adams
    • 1
  • W. Dziobiak
    • 2
  1. 1.Department of Mathematics and Computer ScienceState University of New YorkNew PaltzUSA
  2. 2.Department of MathematicsUniversity of Puerto RicoMayaguezUSA

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