Partial Markov categories are a recent framework for categorical probability theory that provide an abstract account of partial probabilistic computation with updating semantics. In this article, we discuss two order relations on the morphisms of a partial Markov category. In particular, we prove that every partial Markov category is canonically preorder-enriched, recovering several well-known order enrichments. We also demonstrate that the existence of codiagonal maps (comparators) is closely related to order properties of partial Markov categories. Finally, we introduce a synthetic version of the Cauchy--Schwarz inequality and, from it, we prove that updating increases validity.

翻译：偏马尔可夫范畴是范畴概率论的一个新近框架，为具有更新语义的部分概率计算提供了抽象描述。本文讨论偏马尔可夫范畴中态射上的两种序关系。特别地，我们证明每个偏马尔可夫范畴都具有典范的预序富集结构，由此可恢复若干已知的序富化结果。我们还论证了对角态射（比较器）的存在性与偏马尔可夫范畴的序性质密切相关。最后，我们提出柯西-施瓦茨不等式的合成形式，并由此证明更新操作可提升有效性。