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

• Autorzy: Hans-Jürgen Vogel
• Języki publikacji: EN

Abstrakty

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.

Słowa kluczowe

EN

halfdiagonal-halfterminal-symmetric category; diagonal inversion; partial order relation; subidentity; equation

Kategorie tematyczne

• 08A02: Relational systems, laws of composition
• 08A05: Structure theory
• 18D20: Enriched categories (over closed or monoidal categories)
• 18D10: Monoidal categories (= multiplicative categories), symmetric monoidal categories, braided categories
• 18B10: Category of relations, additive relations

• Wydawca: Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra
• Czasopismo: Discussiones Mathematicae - General Algebra and Applications
• Rocznik: 2001
• Tom: 21
• Numer: 2
• Strony: 139-163

Daty

wydano

2001

otrzymano

2000-12-06

Twórcy

autor

Hans-Jürgen Vogel

• University of Potsdam Institute of Mathematics, PF 60 15 53 D-14415 Potsdam, Germany

Bibliografia

• [1] L. B and H.-J. H, Automaten und Funktoren, Akademie-Verlag, Berlin 1975.
• [2] S. E and G.M. K, Closed categories, 'Proceedings of the Conference on Categorical Algebra (La Jolla, 1965)', Springer-Verlag, New York 1966, 421-562.
• [3] H.-J. H, On partial algebras, Colloquia Mathematica Societatis János Bolyai, 29 ('Universal Algebra, Esztergom (Hungary) 1977'), North-Holland, Amsterdam 1981, 373-412.
• [4] G.M. K, On MacLane's condition for coherence of natural associativities, commutativities, etc, J. Algebra 1 (1964), 397-402.
• [5] S. M, Kategorien, Begriffssprache und mathematische Theorie, Springer-Verlag, Berlin-New York 1972.
• [6] J. S, Über dht-symmetrische Kategorien, Semesterarbeit Päd. Hochschule Köthen, Köthen 1978.
• [7] J. S, Über die Einbettung von dht-symmetrischen Kategorien in die Kategorie der partiellen Abbildungen zwischen Mengen, AdW DDR Berlin, Zentralinstitut f. Mathematik und Mechanik, Preprint P-12/8 (1980).
• [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.

Identyfikatory

DOI

10.7151/dmgaa.1034