Geodesic
On the structure of halfdiagonal-halfterminal-symmetric categories with diagonal inversions
Discussiones Mathematicae. General Algebra and Applications, Tome 21 (2001) no. 2, pp. 139-163

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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.
Keywords: halfdiagonal-halfterminal-symmetric category, diagonal inversion, partial order relation, subidentity, equation 
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Vogel, Hans-Jürgen. On the structure of halfdiagonal-halfterminal-symmetric categories with diagonal inversions. Discussiones Mathematicae. General Algebra and Applications, Tome 21 (2001) no. 2, pp. 139-163. http://geodesic.mathdoc.fr/item/DMGAA_2001_21_2_a1/