S4德·摩根代数的成对斯通空间
The pairwise Stone space of an S4 De Morgan algebra
AI总结：
本文研究S4德·摩根代数的双拓扑对偶理论，引入PS4D-空间并证明其与S4德·摩根代数范畴对偶等价，还将该对偶应用于德·摩根代数滤子理想的双拓扑刻画及FDE模态扩张的可靠性完备性。
AI中文摘要：
本研究旨在研究配备闭包算子的德·摩根代数（即S4德·摩根代数）的双拓扑对偶理论。我们首先引入成对斯通空间的某些扩张，称之为成对S4德·摩根斯通空间（简称PS4D-空间），这类空间由一个成对斯通空间$X$、一个扭曲连续对合$g\colon X\to X$以及一个自反且传递的二元关系$R\subseteq X\times X$构成。我们首先证明，S4德·摩根代数$A$的素滤子的双拓扑谱$S_0(A)$构成一个PS4D-空间；随后通过构造从$A$到$S_0(A)$的$(\tau_1,\delta_2)$双开闭子集构成的S4德·摩根代数$A_0(S_0(A))$的同构，得到拓扑表示，其中德·摩根对合运算通过$g$定义，闭包算子通过$R$定义。接着我们给出代数实现定理，证明每个PS4D-空间$X$双同胚且关系同构于$A_0(X)$的素滤子的双拓扑谱$S_0(A_0(X))$。通过引入合适的双连续框架态射，我们证明S4德·摩根代数的范畴$\boldsymbol{S4D}$对偶等价于PS4D-空间的范畴$\boldsymbol{PStone_{S4D}}$。作为应用，我们在建立的对偶框架下给出一般德·摩根代数中滤子与理想的双拓扑刻画，以及一阶蕴涵演算FDE的S4型模态扩张的双拓扑可靠性与完备性结果。
英文摘要：
The purpose of this study is to investigate the bitopological duality theory of De Morgan algebras equipped with a closure operator, known as S4 De Morgan algebras. We first introduce certain expansions of pairwise Stone spaces, which we call pairwise S4 De Morgan Stone spaces (henceforth, PS4D-spaces). These consist of a pairwise Stone space $X$ equipped with a twist continuous involution $g\colon X\to X$, as well as a binary relation $R\subseteq X\times X$ that is reflexive and transitive. We first demonstrate that the bitopological spectrum $S_0(A)$ of prime filters of an S4 De Morgan algebra $A$ gives rise to a PS4D-space. A topological representation is then obtained by exhibiting an isomorphism from $A$ to the S4 De Morgan algebra $A_0(S_0(A))$ of $(τ_1,δ_2)$-biclopen subsets of $S_0(A)$ whose operation of De Morgan involution is defined through $g$ and whose closure operator is defined through $R$. We then provide an algebraic realization theorem by showing that every PS4D-space $X$ is bihomeomorphic and relationally isomorphic to the bitopological spectrum $S_0(A_0(X))$ of prime filters of $A_0(X)$. With the introduction of suitable bicontinuous frame morphisms, we show that the category $\mathbf{S4D}$ of S4 De Morgan algebras is dually equivalent to the category $\mathbf{PStone_{S4D}}$ of PS4D-spaces. As an application, we provide bitopological characterizations of filters and ideals in general De Morgan algebras under our established duality as well as bitopological soundness and completeness results for an S4-type modal extension of the calculus FDE of first-degree entailment.
