Implicative algebras have been recently introduced by Miquel in order to provide a unifying notion of model, encompassing the most relevant and used ones, such as realizability (both classical and intuitionistic), and forcing. In this work, we initially approach implicative algebras as a generalization of locales, and we extend several topological-like concepts to the realm of implicative algebras, accompanied by various concrete examples. Then, we shift our focus to viewing implicative algebras as a generalization of partial combinatory algebras. We abstract the notion of a category of assemblies, partition assemblies, and modest sets to arbitrary implicative algebras, and thoroughly investigate their categorical properties and interrelationships.

翻译：蕴含代数是最近由Miquel引入的，旨在提供一种统一的理论模型概念，涵盖了最相关且最常用的模型，如（经典和直觉主义）可实现性与力迫。在本工作中，我们首先将蕴含代数视为局部格的推广，并将若干拓扑学概念扩展到蕴含代数领域，同时配以各种具体实例。随后，我们将重点转向将蕴含代数视为部分组合代数的推广。我们将装配范畴、划分装配范畴以及适度集合的概念抽象至任意蕴含代数中，并深入研究其范畴性质及其相互关联。