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 Hohle–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.

中文翻译：

---

量值差异

摘要 受Scott 的分居关系理论的启发，我们在不借助否定的情况下，建立了一个在对合量子Q 中取值的正相异性理论。证明了具有 Q 值相异性的集合正是一个对称范畴，它富含 Q 的后对角线的 quantaloid 的 subquantaloid。在松散函子的帮助下，研究了 Q 值相异性和 Q 值相似性（相当于 Hohle-Kubiak 意义上的 Q 值等式）之间的相互作用。特别是，如果 Q 是具有厄密和循环二元化元素的 Girard 量子数，则表明相似性和不同性是可以相互定义的。