PL EN


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
2003 | 23 | 2 | 163-212
Tytuł artykułu

Adjointness between theories and strict theories

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
The categorical concept of a theory for algebras of a given type was foundet by Lawvere in 1963 (see [8]). Hoehnke extended this concept to partial heterogenous algebras in 1976 (see [5]). A partial theory is a dhts-category such that the object class forms a free algebra of type (2,0,0) freely generated by a nonempty set J in the variety determined by the identities ox ≈ o and xo ≈ o, where o and i are the elements selected by the 0-ary operation symbols.
If the object class of a dhts-category forms even a monoid with unit element I and zero element O, then one has a strict partial theory.
In this paper is shown that every J-sorted partial theory corresponds in a natural manner to a J-sorted strict partial theory via a strongly d-monoidal functor. Moreover, there is a pair of adjoint functors between the category of all J-sorted theories and the category of all corresponding J-sorted strict theories.
This investigation needs an axiomatic characterization of the fundamental properties of the category Par of all partial function between arbitrary sets and this characterization leads to the concept of dhts- and dhth∇s-categories, respectively (see [5], [11], [13]).
Twórcy
  • Institute of Mathematics, University of Potsdam, PF 60 15 53, D-14415 Potsdam, Germany
Bibliografia
  • [1] G. Birkhoff and J.D. Lipson, Heterogeneous algebras, J. Combinat. Theory 8 (1970), 115-133.
  • [2] L. Budach and H.-J. Hoehnke, Automaten und Funktoren, Akademie Verlag, Berlin 1975 (the part: 'Allgemeine Algebra der Automaten', by H.-J. Hoehnke).
  • [3] S. Eilenberg and G.M. Kelly, Closed categories, Proc. Conf. Categorical Algebra (La Jolla, 1965), Springer, New York 1966, 421-562.
  • [4] P.J. Higgins, Algebras with a scheme of operators, Math. Nachr. 27 (1963), 115-132.
  • [5] H.-J. Hoehnke, On partial algebras, Colloq. Soc. J. Bolyai, Vol. 29, 'Universal Algebra; Esztergom (Hungary) 1977', North-Holland, Amsterdam, 1981, 373-412.
  • [6] G.M. Kelly, On MacLane's conditions for coherence of natural associativities, commutativities, etc, J. Algebra 4 (1964), 397-402.
  • [7] G.M. Kelly and S. MacLane, Coherence in closed categories, + Erratum, Pure Appl. Algebra 1 (1971), 97-140, 219.
  • [8] F.W. Lawvere, Functorial semantics of algebraic theories, Proc. Nat. Acad. Sci. U.S.A. 50 (1963), 869-872.
  • [9] S. MacLane, Natural associativity and commutativity, Rice Univ. Studies 49 (1963), no. 4, 28-46.
  • [10] J. Schreckenberger, Über die zu monoidalen Kategorien adjungierten Kronecker-Kategorien, Dissertation (A), Päd. Hochschule, Potsdam 1978.
  • [11] J. Schreckenberger, Über die Einbettung von dht-symmetrischen Kategorien in die Kategorie der partiellen Abbildungen zwischen Mengen, Preprint P-12/80, Zentralinst. f. Math., Akad. d. Wiss. d. DDR, Berlin 1980.
  • [12] J. Schreckenberger, Zur Theorie der dht-symmetrischen Kategorien, Disseration (B), Päd. Hochschule Potsdam, Math.-Naturwiss. Fak., Potsdam 1984.
  • [13] H.-J. Vogel, Eine kategorientheoretische Sprache zur Beschreibung von Birkhoff-Algebren, Report R-Math-06/84, Inst. f. Math., Akad. d. Wiss. d. DDR, Berlin 1984.
  • [14] H.-J. Vogel, On functors between dht∇ -symmetric categories, Discuss. Math.-Algebra & Stochastic Methods, 18 (1998), 131-147.
  • [15] H.-J. Vogel, On properties of dht∇ -symmetric categories, Contributions to General Algebra 11 (1999), 211-223.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-doi-10_7151_dmgaa_1071
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.