分割预订的价格
Syntax for Split Preorders
摘要：
分裂前序是两个集合的不交并上的一个前序关系，这两个集合在合成分裂前序时充当源和目标。本文通过生成元和方程给出了SplPre范畴，其箭头是两个有限序数的不交并上的分裂前序。对于SplPre的子范畴Gen和范畴Rel也是如此，Gen的箭头是等价关系，Rel的箭头是有限序数之间的二元关系，并且Rel在SplPre中有一个保持合成的映射的同构像，但不保持恒等箭头。先前已经表明，SplPre和Gen在Brauer风格的Rel中具有同构表示。本文对Gen和Rel的句法表示，展示了Gen特殊的Frobenius代数结构和Rel特殊的双代数结构，后者是建立在SplPre中Gen和Rel的双代数结构之上的。这指向代数建模的各种类别的动机的逻辑，和相关的类别，其中一个可以建立连贯性方面的Rel和根。它也揭示了之间的关系概念的弗罗贝纽斯代数和双代数。语法表示的完备性通过规范形式证明，SplPre和Gen的规范形式在某种意义上正交于Rel的无组合（即无切割）规范形式。本文最后表明，假设的代数结构的SplPre，根和Rel不能扩展新的方程，而不会陷入平凡。
英文摘要：
A split preorder is a preordering relation on the disjoint union of two sets, which function as source and target when one composes split preorders. The paper presents by generators and equations the category SplPre, whose arrows are the split preorders on the disjoint union of two finite ordinals. The same is done for the subcategory Gen of SplPre, whose arrows are equivalence relations, and for the category Rel, whose arrows are the binary relations between finite ordinals, and which has an isomorphic image within SplPre by a map that preserves composition, but not identity arrows. It was shown previously that SplPre and Gen have an isomorphic representation in Rel in the style of Brauer. The syntactical presentation of Gen and Rel in this paper exhibits the particular Frobenius algebra structure of Gen and the particular bialgebraic structure of Rel, the latter structure being built upon the former structure in SplPre. This points towards algebraic modelling of various categories motivated by logic, and related categories, for which one can establish coherence with respect Rel and Gen. It also sheds light on the relationship between the notions of Frobenius algebra and bialgebra. The completeness of the syntactical presentations is proved via normal forms, with the normal form for SplPre and Gen being in some sense orthogonal to the composition-free, i.e. cut-free, normal form for Rel. The paper ends by showing that the assumptions for the algebraic structures of SplPre, Gen and Rel cannot be extended with new equations without falling into triviality.
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