格值关系范畴中的箭头运算
Arrow Operations in Categories of Lattice-valued Relations
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中文总结 AI 辅助
本文将箭头 allegory 的框架扩展到使用不同真值格的关系，定义了三个对应不同情况的具体 allegory，并研究了它们及相关范畴定义。
中文摘要 AI 辅助
箭头 allegory（寓言）为处理格值关系提供了便捷的抽象框架，更准确地说，是使用给定Heyting代数元素作为真值的关系。箭头 allegory 的一个特征是，给定箭头 allegory 中的所有关系都使用同一个Heyting代数$\boldsymbol{\textit{H}}$。本文将该方法扩展到以下情况：不同对象间的关系可使用不同的真值格，甚至每对关系可使用不同的真值格。为此，我们定义了三个具体的 allegory：$\boldsymbol{\textit{Rel}}(\boldsymbol{\textit{H}})$、$\boldsymbol{\textit{Rel}}^u(\boldsymbol{\textit{H}})$和$\boldsymbol{\textit{H}}\boldsymbol{\textit{-Rel}}$，其中后一个 allegory 是前一个的全子 allegory。这三个 allegory 分别对应上述三种不同情况，特别是$\boldsymbol{\textit{H}}\boldsymbol{\textit{-Rel}}$是箭头范畴的标准示例。我们对这些 allegory 进行了研究，并为这些结构提供了合适的范畴定义。
英文摘要
Arrow allegories provide a convenient abstract framework to work with lattice-valued relations, or more precisely, relations that use the elements of a given Heyting algebra as truth values. One characteristic of arrow allegories is that all relations of the given arrow allegory use the same Heyting algebra ${\mathcal H}$. In this paper we want to extend this approach to allegories where relations between different objects may use different lattices of truth values and even further to relations that use a different lattice of truth values for every pair in the relation. Therefore, we define three concrete allegories, $\mathrm{Rel}({\mathcal H})$, $\mathrm{Rel}^u({\mathcal H})$ and ${\mathcal H}{\rm-Rel}$, where the allegory listed later is a full suballegory of the previous ones. These three allegories capture the three different situations mentioned above. In particular, ${\mathcal H}{\rm-Rel}$ is the standard example of an arrow category. We investigate these allegories and provide suitable categorical definitions for these structures.
