We introduce the notion of implicative algebra, a simple algebraic structure intended to factorize the model-theoretic constructions underlying forcing and realizability (both in intuitionistic and classical logic). The salient feature of this structure is that its elements can be seen both as truth values and as (generalized) realizers, thus blurring the frontier between proofs and types. We show that each implicative algebra induces a (Set-based) tripos, using a construction that is reminiscent from the construction of a realizability tripos from a partial combinatory algebra. Relating this construction with the corresponding constructions in forcing and realizability, we conclude that the class of implicative triposes encompasses all forcing triposes (both intuitionistic and classical), all classical realizability triposes (in the sense of Krivine), and all intuitionistic realizability triposes built from partial combinatory algebras.

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隐含代数：可实现性和强迫性的新基础

我们介绍了隐含代数的概念，这是一种简单的代数结构，旨在分解强迫和可实现性（在直觉逻辑和经典逻辑中）的模型理论结构。这种结构的显着特征是它的元素既可以被视为真值，也可以被视为（广义）实现者，从而模糊了证明和类型之间的边界。我们证明了每个隐含代数都诱导出一个 (放-based) tripos，使用的结构让人想起从部分组合代数构建可实现性 tripos。将此结构与强制和可实现性中的相应结构联系起来，我们得出结论，隐含三元组包括所有强制三元组（直觉和经典）、所有经典可实现性三元组（在 Krivine 意义上）和所有直觉可实现性三元组部分组合代数。