Advertisement

Applied Categorical Structures

, Volume 1, Issue 4, pp 423–440 | Cite as

The algebraic theory of order

  • Hans-E. Porst
Article

Abstract

Partially ordered sets are described in terms of partial operations with equationally defined domains and equations, thus the categoryPOS of posets is represented as a one-sorted essentially algebraic category in the sense of Freyd [7] which, in this case even can be fully embedded into a non-trivial variety. This is achieved by using the relation of a poset rather than its underlying set as the carrier set of the algebraic structure. Essentially equational descriptions of somePOS-based algebraic structures are given, and an equational characterization of Galois connections is obtained.

Mathematics Subject Classifications (1991)

Primary 06A06 06A15 06F99 Secondary 08A55 18B35 

Key words

Essentially equational theory essentially algebraic category order and preorder ordered algebras 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    J. Adámek, H. Herrlich, and J. Rosický: Essentially equational categories,Cahiers Topol. Géom. Différentielles Catégoriques XXIX (1988), 175–192.Google Scholar
  2. 2.
    J. Adámek, H. Herrlich, and G.E. Strecker:Abstract and Concrete Categories, Wiley Interscience, New York, 1990.Google Scholar
  3. 3.
    J. Adámek, H. Herrlich, and W. Tholen: Monadic decompositions,J. Pure Appl. Algebra 59 (1989), 111–123.Google Scholar
  4. 4.
    M. Barr: HSP type theorems in the category of posets, in:Mathematical Foundations of Programming Semantics, LNCS598, Springer, Berlin - New York 1992, 221–234.Google Scholar
  5. 5.
    M. Barr and C. Wells:Toposes, Triples and Theories, Springer, Berlin - New York 1985.Google Scholar
  6. 6.
    R. Börger and W. Tholen: Strong, regular and dense generators,Cahiers Topol. Géom. Différentielles Catégoriques XXXII (1991), 257–276.Google Scholar
  7. 7.
    P. Freyd: Aspects of topoi,Bull. Austral. Math. Soc. 7 (1972), 1–76.Google Scholar
  8. 8.
    P. Gabriel and F. Ulmer:Lokal präsentierbare Kategorien, LNM221, Springer, Berlin - New York 1971.Google Scholar
  9. 9.
    H. Herrlich, Regular categories and regular functors,Canad. J. Math. XXVI (1974), 709–720Google Scholar
  10. 10.
    H. Herrlich, Essentially algebraic categories,Quaestiones Math. 9 (1986) 245–262Google Scholar
  11. 11.
    S. Mac Lane,Categories for the Working Mathematician, GTM5, Springer, Berlin - New York 1971Google Scholar
  12. 12.
    E. Makai jun., Automorphisms and Full Embeddings of Categories in Algebra and Topology, in:Category Theory at Work, Heldermann-Verlag, Berlin 1991, 217–260Google Scholar
  13. 13.
    E.G. Manes,Algebraic Theories, GTM26, Springer, Berlin - New York 1976Google Scholar
  14. 14.
    L.D. Nel, Initially structured categories,Can. J. Math. XXVII (1975), 1361–1377Google Scholar
  15. 15.
    H.-E. Porst, T-regular functors, in:Categorical Topology, Heldermann-Verlag, Berlin 1984, 425–440Google Scholar
  16. 16.
    H.-E. Porst, What is concrete equivalence?Seminarberichte Mathematik 44, Fernuniversität Hagen (1992), 312–321Google Scholar
  17. 17.
    H.-E. Porst, The Linton Theorem revisited,Cahiers Topol. Géom. Différentielles Catégoriques XXXIV (1993), 229–238.Google Scholar
  18. 18.
    H. Reichel,Structural Induction on Partial Algebras, Akademie Verlag, Berlin 1984Google Scholar
  19. 19.
    M. Sekanina, Realisation of ordered sets by means of universal algebras, especially by semigroups, in:Theory of Sets and Topology (in honour of F. Hausdorff), VEB Deutsch. Verlag Wiss. Berlin, 1972, 455–466Google Scholar

Copyright information

© Kluwer Academic Publishers 1993

Authors and Affiliations

  • Hans-E. Porst
    • 1
  1. 1.Fachbereich Mathematik und InformatikUniversität BremenBremenGermany

Personalised recommendations