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

Hans-Jürgen Vogel

ArticleOriginal scientific text

Title

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

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Affiliations

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

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 \to A ⨂ A | A \in | R e l ∣)

of diagonal morphisms, a family

t = (t_{A} : A \to I | A \in | R e l ∣)

of terminal morphisms, and a family

\nabla = (\nabla_{A} : A ⨂ A \to A | A \in | R e l ∣)

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.

Keywords

ENG

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

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Title

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

Affiliations

Abstract

Keywords

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