Download PDF - Retracts and Q-independenceAll rights reserved. Copyright exceptions and limitations apply

ArticleOriginal scientific text

Title

Retracts and Q-independence

Authors 1

Affiliations

  1. Opole University of Technology, Waryńskiego 4, 45-047 Opole, Poland

Abstract

A non-empty set X of a carrier A of an algebra A is called Q-independent if the equality of two term functions f and g of the algebra A on any finite system of elements a₁,a₂,...,aₙ of X implies f(p(a₁),p(a₂),...,p(aₙ)) = g(p(a₁),p(a₂),...,p(aₙ)) for any mapping p ∈ Q. An algebra B is a retract of A if B is the image of a retraction (i.e. of an idempotent endomorphism of B). We investigate Q-independent subsets of algebras which have a retraction in their set of term functions.

Keywords

ENG
general algebra, term function, Q-independence, M, I, S, S₀, A₁, G-independence, t-independence, retraction, retract, Stone algebra, skeleton and set of dense element of Stone algebra, Glivenko congruence

Bibliography

  1. R. Balbes and Ph. Dwinger, Distributive Lattices, Univ. Missouri Press, Columbia, Miss. 1974.
  2. S. Burris and H.P. Sankappanavar, A Course in Universal Algebra, The Millennium Edition 2000.
  3. C.C. Chen and G. Grätzer, Stone lattices. I: Construction theorems, Can. J. Math. 21 (1969), 884-894.
  4. A. Chwastyk and K. Głazek, Remarks on Q-independence of Stone algebras, Math. Slovaca 56 (2) (2006), 181-197.
  5. K. Głazek, General notions of independence, pp. 112-128 in 'Advances in Algebra', World Scientific, Singapore 2003.
  6. K. Głazek, Independence with respect to family of mappings in abstract algebras, Dissertationes Math. 81 (1971).
  7. K. Głazek, Some old and new problems in the independence theory, Coll. Math. 42 (1979), 127-189.
  8. K. Głazek and S. Niwczyk, A new perspective on Q-independence, General Algebra and Applications, Shaker Verlag, Aacken 2000.
  9. K. Golema-Hartman, Exchange property and t-independence, Coll. Math. 36 (1976), 181-186.
  10. G. Grätzer, A new notion of independence in universal algebras, Colloq. Math. 17 (1967), 225-234.
  11. G. Grätzer, General Lattice Theory, Academic Press, New York 1978.
  12. G. Grätzer, Universal Algebra, second edition, Springer-Verlag, New York 1979.
  13. E. Marczewski, A general scheme of the notions of independence in mathematics, Bull. Acad. Polon. Sci. (Ser. Sci. Math. Astr. Phys.) 6 (1958), 731-738.
  14. E. Marczewski, Concerning the independence in lattices, Colloq. Math. 10 (1963), 21-23.
  15. E. Marczewski, Independence in algebras of sets and Boolean algebras, Fund. Math. 48 (1960), 135-145.
  16. E. Marczewski, Independance with respect to a family of mappings, Colloq. Math. 20 (1968), 11-17.
  17. J. Płonka and W. Poguntke, T-independence in distributive lattices, Colloq. Math. 36 (1976), 171-175.
  18. J. Schmidt, Eine algebraische Äquivalenz zum Auswahlaxiom, Fund. Math. 50 (1962), 485-496.
  19. S. Świerczkowski, Topologies in free algebras, Proc. London Math. Soc. 14 (3) (1964), 566-576.
  20. G. Szász, Marczewski independence in lattices and semilattices, Colloq. Math. 10 (1963), 15-23.
Discussiones Mathematicae - General Algebra and Applications
2007/27/2
Export to PBN, Opens in new tab
Pages:
235-243
Main language of publication
English
Received
2006-05-15
Accepted
2006-08-24
Published
2007

DOI: 10.7151/dmgaa.1128

Exact and natural sciences