有界蕴涵代数的单子和柱形扩张
Monadic and cylindric expansions of bounded implication algebras
AI总结：
本文在有界蕴涵代数上引入单子和柱形算子，证明单子蕴涵代数范畴与单子布尔代数范畴同构，以及I维柱形蕴涵代数范畴与I维柱形布尔代数范畴同构，并应用该同构给出I维柱形蕴涵代数的谱对偶结果。
摘要：
Abbott引入了蕴涵代数作为经典命题演算中布尔蕴涵运算的代数模型。本文通过引入单子和柱形蕴涵代数，研究有界蕴涵代数上的额外算子和常量。证明了单子蕴涵代数范畴$\mathbf{MIA}$与单子布尔代数范畴$\mathbf{MBA}$同构，并且$I$维柱形蕴涵代数范畴$\mathbf{CIA}$与$I$维柱形布尔代数范畴$\mathbf{CBA}$同构。作为所得范畴同构的应用，我们沿着Bezhanishvili和Holliday对布尔代数的谱对偶以及McDonald将其对偶扩展到单子和$I$维柱形布尔代数的思路，给出了$I$维柱形蕴涵代数的谱对偶结果。
英文摘要：
Implication algebras were introduced by Abbott as algebraic models of the operation of Boolean implication in the classical propositional calculus. In this work, we study additional operators and constants on bounded implication algebras by introducing monadic and cylindric implication algebras. It is demonstrated that the category $\mathbf{MIA}$ of monadic implication algebras is isomorphic to the category $\mathbf{MBA}$ of monadic Boolean algebras and moreover, that the category $\mathbf{CIA}$ of $I$-dimensional cylindric implication algebras is isomorphic to the category $\mathbf{CBA}$ of $I$-dimensional cylindric Boolean algebras. As an application of the obtained categorical isomorphisms, we provide spectral duality results for $I$-dimensional cylindric implication algebras along the lines of Bezhanishvili and Holliday's spectral duality for Boolean algebras combined with McDonald's extension of their duality to monadic and $I$-dimensional cylindric Boolean algebras.
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