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Discussiones Mathematicae General Algebra and Applications 21(2) (2001)139-163
DOI: https://doi.org/10.7151/dmgaa.1034

ON THE STRUCTURE OF HALFDIAGONAL-HALFTERMINAL-SYMMETRIC CATEGORIES WITH DIAGONAL INVERSIONS

Hans-Jürgen Vogel

University of Potsdam Institute of Mathematics
PF 60 15 53 D-14415 Potsdam, Germany
e-mail: vogel@rz.uni-potsdam.de

Dedicated to Hans-Jürgen Hoehnke on the occasion of his 75th birthday.

Abstract

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 ÄA  | A Î |Rel|) of diagonal morphisms, a family t = (t_A: A ® I  | A Î |Rel|) of terminal morphisms, and a family Ń = (Ń_A: A ÄA ® A  | A Î |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 (hdhtŃs-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 hdhtŃs-categories consisting of identities only. Therefore, the class of all small hdhtŃ-symmetric categories (interpreted as hetrogeneous algebras of a certain type) forms a variety and hence there are free theories for relational structures.

Keywords: halfdiagonal-halfterminal-symmetric category, diagonal inversion, partial order relation, subidentity, equation.

2000 AMS Subject Classification: 18D10, 18B10, 18D20, 08A05, 08A02.

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Received 6 December 2000