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2001 | 21 | 2 | 139-163
Tytuł artykułu

On the structure of halfdiagonal-halfterminal-symmetric categories with diagonal inversions

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EN
The category of all binary relations between arbitrary sets turns out to be a certain symmetric monoidal category Rel with an additional structure characterized by a family $d = (d_{A}: A → A⨂ A | A ∈ |Rel|)$ of diagonal morphisms, a family $t = (t_{A}: A → I | A ∈ |Rel|)$ of terminal morphisms, and a family $∇ = (∇_{A}: A ⨂ A → A | A ∈ |Rel|)$ of diagonal inversions having certain properties. Using this properties in [11] was given a system of axioms which characterizes the abstract concept of a halfdiagonal-halfterminal-symmetric monoidal category with diagonal inversions (hdht∇s-category). Besides of certain identities this system of axioms contains two identical implications. In this paper is shown that there is an equivalent characterizing system of axioms for hdht∇s-categories consisting of identities only. Therefore, the class of all small hdht∇-symmetric categories (interpreted as hetrogeneous algebras of a certain type) forms a variety and hence there are free theories for relational structures.
Twórcy
  • University of Potsdam Institute of Mathematics, PF 60 15 53 D-14415 Potsdam, Germany
Bibliografia
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  • [8] J. S, Zur Theorie der dht-symmetrischen Kategorien, Dissertation (B), eingereicht an der Math.-Naturwiss. Fak. d. Wiss. Rates d. Pädag. Hochschule 'Karl Liebknecht', Potsdam 1984.
  • [9] J. S, Zur Axiomatik von Kategorien partieller Morphismen, Beiträge Algebra Geom. (Halle/Saale) 24 (1987), 83-98.
  • [10] H.-J. V, Eine kategorientheoretische Sprache zur Beschreibung von Birkhoff-Algebren, AdW DDR, Institut f. Mathematik, Report R-MATH-06/84, Berlin (1984).
  • [11] H.-J. V, Relations as morphisms of a certain monoidal category, 'General Algebra and Applications in Discrete Mathematics', Shaker Verlag, Aachen 1997, 205-217.
  • [12] H.-J. V, On functors between dhtŃ-symmetric categories, Discuss. Math. - Algebra & Stochastic Methods 18 (1998), 131-147.
  • [13] H.-J. V, On Properties of dhtŃ-symmetric categories, Contributions to General Algebra 11 (1999), 211-223.
  • [14] H.-J. V, Halfdiagonal-halfterminal-symmetric monoidal categories with inversions, 'General Algebra and Discrete Mathematics', Shaker Verlag, Aachen 1999, 189-204.
  • [15] H.-J. V, On morphisms between prtial algebras, 'Algebras and Combinatorics. An International Congress, ICAC '97, Hong Kong', Springer-Verlag, Singapore 1999, 427-453.
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bwmeta1.element.bwnjournal-article-doi-10_7151_dmgaa_1034
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