DM-GAA

ISSN 1509-9415 (print version)

ISSN 2084-0373 (electronic version)

https://doi.org/10.7151/dmgaa

Discussiones Mathematicae - General Algebra and Applications

Cite Score: 0.4

SJR: 0.203

SNIP: 0.562

MCQ: 0.12

Index Copernicus: 121.02

Discussiones Mathematicae - General Algebra and Applications

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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 = (dA: A ® A ÄA  | A Î |Rel|) of diagonal morphisms, a family t = (tA: A ® I  | A Î |Rel|) of terminal morphisms, and a family Ń = (ŃA: 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.

References

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Received 6 December 2000


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