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 of diagonal morphisms, a family of terminal morphisms, and a family 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgaa_1034,
author = {Hans-J\"urgen Vogel},
title = {On the structure of halfdiagonal-halfterminal-symmetric categories with diagonal inversions},
journal = {Discussiones Mathematicae - General Algebra and Applications},
volume = {21},
year = {2001},
pages = {139-163},
zbl = {0998.18003},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgaa_1034}
}
Hans-Jürgen Vogel. On the structure of halfdiagonal-halfterminal-symmetric categories with diagonal inversions. Discussiones Mathematicae - General Algebra and Applications, Tome 21 (2001) pp. 139-163. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgaa_1034/
[000] [1] L. B and H.-J. H, Automaten und Funktoren, Akademie-Verlag, Berlin 1975.
[001] [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.
[002] [3] H.-J. H, On partial algebras, Colloquia Mathematica Societatis János Bolyai, 29 ('Universal Algebra, Esztergom (Hungary) 1977'), North-Holland, Amsterdam 1981, 373-412.
[003] [4] G.M. K, On MacLane's condition for coherence of natural associativities, commutativities, etc, J. Algebra 1 (1964), 397-402. | Zbl 0246.18008
[004] [5] S. M, Kategorien, Begriffssprache und mathematische Theorie, Springer-Verlag, Berlin-New York 1972.
[005] [6] J. S, Über dht-symmetrische Kategorien, Semesterarbeit Päd. Hochschule Köthen, Köthen 1978.
[006] [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). | Zbl 0442.18002
[007] [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.
[008] [9] J. S, Zur Axiomatik von Kategorien partieller Morphismen, Beiträge Algebra Geom. (Halle/Saale) 24 (1987), 83-98.
[009] [10] H.-J. V, Eine kategorientheoretische Sprache zur Beschreibung von Birkhoff-Algebren, AdW DDR, Institut f. Mathematik, Report R-MATH-06/84, Berlin (1984).
[010] [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.
[011] [12] H.-J. V, On functors between dhtŃ-symmetric categories, Discuss. Math. - Algebra & Stochastic Methods 18 (1998), 131-147. | Zbl 0921.18005
[012] [13] H.-J. V, On Properties of dhtŃ-symmetric categories, Contributions to General Algebra 11 (1999), 211-223.
[013] [14] H.-J. V, Halfdiagonal-halfterminal-symmetric monoidal categories with inversions, 'General Algebra and Discrete Mathematics', Shaker Verlag, Aachen 1999, 189-204.
[014] [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. | Zbl 0968.08003