扩展贝尔纳普2：深入的对偶范畴
Expanding Belnap 2: the dual category in depth
摘要：
双格是由N. D. 1977年，贝尔纳普发表了一篇题为《计算机应该如何思考》的论文。优先默认bilattices不仅包括贝尔纳普的四个值，为“真”（$t$），“假”（$f$），“矛盾”（$\top$）和“无信息”（$\bot$），但也索引家庭的默认值，同时建模程度的知识和真理。优先缺省双格在包括人工智能在内的许多领域都有应用。在我们的配套文件中，我们介绍了一个新的家庭的优先默认双格，$\mathbf J_n$，$n \in ω$，与$\mathbf J_0$是贝尔纳普的开创性的例子。给出了由$\mathbf J_n$生成的簇$\mathcal V_n$的一个对偶，其中对偶范畴$\mathcal X_n$的对象是多序拓扑结构.本文对对偶范畴进行了深入的研究。我们给出范畴$\mathcal X_n$的一个公理化，并证明它同构于单序拓扑结构范畴$\mathcal Y_n$. $\mathcal Y_n$的对象是具有连续收缩的Priestley空间，其中序具有自然的排序。我们展示了如何通过$\mathcal Y_n$中的对偶来构造$\mathcal V_n$中的代数的底层有界分配格的Priestley对偶;作为应用，我们证明了自由代数$\mathbf F_{\mathcal V_n}(1)$的大小由$n$中的多项式给出。
英文摘要：
Bilattices, which provide an algebraic tool for simultaneously modelling knowledge and truth, were introduced by N.D. Belnap in a 1977 paper entitled 'How a computer should think'. Prioritised default bilattices include not only Belnap's four values, for `true' ($t$), `false'($f$), `contradiction' ($\top$) and `no information' ($\bot$), but also indexed families of default values for simultaneously modelling degrees of knowledge and truth. Prioritised default bilattices have applications in a number of areas including artificial intelligence. In our companion paper, we introduced a new family of prioritised default bilattices, $\mathbf J_n$, for $n \in ω$, with $\mathbf J_0$ being Belnap's seminal example. We gave a duality for the variety $\mathcal V_n$ generated by $\mathbf J_n$, with the objects of the dual category $\mathcal X_n$ being multi-sorted topological structures. Here we study the dual category in depth. We give an axiomatisation of the category $\mathcal X_n$ and show that it is isomorphic to a category $\mathcal Y_n$ of single-sorted topological structures. The objects of $\mathcal Y_n$ are Priestley spaces endowed with a continuous retraction in which the order has a natural ranking. We show how to construct the Priestley dual of the underlying bounded distributive lattice of an algebra in $\mathcal V_n$ via its dual in $\mathcal Y_n$; as an application we show that the size of the free algebra $\mathbf F_{\mathcal V_n}(1)$ is given by a polynomial in $n$ of degree $6$.
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