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ArticleOriginal scientific text

Title

Duality for some free modes

Authors 1, 1, 2

Affiliations

  1. Faculty of Mathematics and Information Sciences, Warsaw University of Technology, 00-661 Warsaw, Poland
  2. Department of Mathematics, Iowa State University, Ames, Iowa 50011, U.S.A.

Abstract

The paper establishes a duality between a category of free subreducts of affine spaces and a corresponding category of generalized hypercubes with constants. This duality yields many others, in particular a duality between the category of (finitely generated) free barycentric algebras (simplices of real affine spaces) and a corresponding category of hypercubes with constants.

Keywords

ENG
duality, modes, affine spaces and their subreducts, barycentric algebras, convex sets, simplices, hypercubes

Bibliography

  1. R. Balbes and P. Dwinger, Distributive Lattices, University of Missouri Press, Columbia, MO, 1974.
  2. D.M. Clark and B.A. Davey, Natural Dualities for the Working Algebraists, Cambridge University Press, Cambridge 1998.
  3. B. Csákány, Varieties of affine modules, Acta Sci Math. 37 (1975), 3-10.
  4. B.A. Davey, Duality theory on ten dollars a day, 'Algebras and Orders', Kluwer Acad. Publ. 1993, 71-111.
  5. B.A. Davey and R.W. Quackenbush, Bookkeeping duality for paraprimal algebras, Contributions to General Algebra 9 (1995), 19-26.
  6. B.A. Davey and H. Werner, Dualities and equivalences for varieties of algebras, Colloq. Math. Soc. J. Bolyai 33 (1980), 101-275.
  7. G. Grätzer, Universal Algebra, Springer-Verlag, Berlin 1979.
  8. K.H. Hofmann, M. Mislove and A. Stralka, The Pontryagin Duality of Compact 0-Dimensional Semilattices and its Applications, Springer-Verlag,Berlin 1974.
  9. J. Jezek and T. Kepka, Free commutative idempotent abelian groupoids and quasigroups, Acta Univ. Carolin. Math. Phys. 17 (1976), 13-19.
  10. J. Jezek and T. Kepka, Medial Groupoids, Academia, Praha 1983.
  11. S. MacLane, Categories for the Working Mathematician, Springer-Verlag, Berlin 1971.
  12. A. I. Mal'cev, Algebraic Systems, Springer-Verlag, New York 1973.
  13. W.D. Neumann, On the quasivariety of convex subsets of affine spaces, Arch. Math. 21 (1970), 11-16.
  14. K. Pszczoła, Duality for affine spaces over finite fields, Contributions to General Algebra 13 (2001), 285-293.
  15. K.J. Pszczoła, A.B. Romanowska and J.D.H. Smith, Duality for quadrilaterals, Contribution to General Algebra, to appear.
  16. A.B. Romanowska, Barycentric algebras, 'General Algebra and Applications', Shaker Verlag, Aachen 2000, 167-181.
  17. A.B. Romanowska and J.D.H. Smith, Modal Theory, Heldermann-Verlag, Berlin 1985.
  18. A.B. Romanowska and J.D.H. Smith, Semilattice-based dualities, Studia Logica 56 (1996), 225-261.
  19. A.B. Romanowska and J.D.H. Smith, Duality for semilattice representations, J. Pure Appl. Algebra 115 (1997), 289-308.
  20. A.B. Romanowska and J.D.H. Smith, Embedding sums of cancellatice modes into functorial sums of affine spaces, 'Unsolved Problems on Mathematics for the 21st Century, a Tribute to Kiyoshi Iseki's 80th Birthday', IOS Press, Amsterdam 2001, 127-139.
  21. A.B. Romanowska and J.D.H. Smith, Modes, World Scientific, Singapore 2002.
  22. A.B. Romanowska and J.D.H. Smith, Poset extensions, convex sets, and semilattice presentations, preprint, 2002.
  23. J.D.H. Smith and A. B. Romanowska, Post-Modern Algebra, Wiley, New York, NY, 1999.
Discussiones Mathematicae - General Algebra and Applications
2003/23/1
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Pages:
45-61
Main language of publication
English
Received
2003-03-04
Published
2003

DOI: 10.7151/dmgaa.1063

Exact and natural sciences