Abstract
Inspired by the theory of apartness relations of Scott, we establish a positive theory of dissimilarity valued in an involutive quantale Q without the aid of negation. It is demonstrated that a set equipped with a Q-valued dissimilarity is precisely a symmetric category enriched in a subquantaloid of the quantaloid of back diagonals of Q. Interactions between Q-valued dissimilarities and Q-valued similarities (which are equivalent to Q-valued equalities in the sense of Höhle–Kubiak) are investigated with the help of lax functors. In particular, it is shown that similarities and dissimilarities are interdefinable if Q is a Girard quantale with a hermitian and cyclic dualizing element.
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