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Quantale-valued dissimilarity
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DOI：10.1016/j.fss.2020.01.013.png)
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Inspired by the theory of apartness relations of Scott, we establish a positive theory of dissimilarity valued in an involutive quantale Qwithout 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 Hohle-Kubiak) are investigated with the help of lax functors. In particular, it is shown that similarities and dissimilarities are interdefinable if Qis a Girard quantale with a hermitian and cyclic dualizing element. (c) 2020 Elsevier B.V. All rights reserved.
Keyword:
Dissimilarity
Similarity
Quantale
Back diagonal
Diagonal
Quantaloid
Girard quantale
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