某些模态和剩余群胚扩张的De Morgan代数的谱对偶
Spectral duality for some modal and residuated groupoid expansions of De Morgan algebras
AI总结：
研究De Morgan代数及其扩张（如S4 De Morgan代数和De Morgan群胚）的谱对偶理论，通过调整Bimbó和Urquhart的Priestley对偶，在谱空间与Priestley空间同构下获得对偶结果。
摘要：
Stone证明了有界分配格范畴$\mathbf{DLATT_{0,1}}$与谱空间范畴$\mathbf{Spec}$对偶等价，Priestley证明了$\mathbf{DLatt_{0,1}}$与Priestley空间范畴$\mathbf{Priest}$对偶等价，从而$\mathbf{Spec}$与$\mathbf{Priest}$等价。Cornish加强了这一结果，证明$\mathbf{Spec}$与$\mathbf{Priest}$实际上同构。本研究探讨了某些有界分配格序代数（称为De Morgan代数）的各种格扩张的对偶理论。特别地，我们获得了带有闭包算子的De Morgan代数范畴$\mathbf{S4DM}$（称为S4 De Morgan代数）以及De Morgan群胚范畴$\mathbf{DMGrp}$的谱对偶结果。这是通过适当调整Bimbó对一般De Morgan代数的Priestley式对偶以及Urquhart对相关代数的Priestley式对偶，并在$\mathbf{Priest}$与$\mathbf{Spec}$同构下实现的。
英文摘要：
Stone demonstrated that the category $\mathbf{DLATT_{0,1}}$ of bounded distributive lattices is dually equivalent to the category $\mathbf{Spec}$ of spectral spaces and Priestley showed that $\mathbf{DLatt_{0,1}}$ is dually equivalent to the category $\mathbf{Priest}$ of Priestley spaces so that $\mathbf{Spec}$ is equivalent $\mathbf{Priest}$. Cornish strengthened this by showing that $\mathbf{Spec}$ and $\mathbf{Priest}$ are in fact isomorphic. In this study, we investigate the duality theory of various lattice expansions of certain bounded distributive lattice-ordered algebras, known as De Morgan algebras. In particular we obtain spectral duality results for the category $\mathbf{S4DM}$ of De Morgan algebras equipped with a closure operator, which we call S4 De Morgan algebras, as well as for the category $\mathbf{DMGrp}$ of De Morgan groupoids. This is achieved by an appropriate adaptation of Bimbó's Priestley-style duality for general De Morgan algebras together with Urquhart's Priestley-style duality for relevance algebras under the isomorphism between $\mathbf{Priest}$ and $\mathbf{Spec}$.
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