Abstract In this paper we analyze some classes of abstract simplicial complexes relying on algebraic models arising from module theory. To this regard, we consider a left-module on a unitary ring and find models of abstract complexes and related set operators having specific regularity properties, which are strictly interrelated to the algebraic properties of both the module and the ring. Next, taking inspiration from the aforementioned models, we carry out our analysis from modules to arbitrary sets. In such a more general perspective, we start with an abstract simplicial complex and an associated set operator. Endowing such a set operator with the corresponding properties obtained in our module instances, we investigate in detail and prove several properties of three subclasses of abstract complexes. More specifically, we provide uniformity conditions in relation to the cardinality of the maximal members of such classes. By means of the notion of OSS-bijection, we prove a correspondence theorem between a subclass of closure operators and one of the aforementioned families of abstract complexes, which is similar to the classic correspondence theorem between closure operators and Moore systems. Next, we show an extension property of a binary relation induced by set systems when they belong to one of the above families. Finally, we provide a representation result in terms of pairings between sets for one of the three classes of abstract simplicial complexes studied in this work.

中文翻译：

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由模理论驱动的一些抽象单纯复形

摘要 在本文中，我们分析了一些依赖于模理论产生的代数模型的抽象单纯复形。在这方面，我们考虑幺正环上的左模，并找到抽象复形模型和具有特定正则性的相关集合算子，它们与模和环的代数性质密切相关。接下来，从上述模型中汲取灵感，我们从模块到任意集合进行分析。从更一般的角度来看，我们从一个抽象单纯复形和一个相关的集合算子开始。赋予这样一个集合运算符在我们的模块实例中获得的相应属性，我们详细研究并证明了抽象复合体的三个子类的几个属性。进一步来说，我们提供与此类类的最大成员的基数相关的一致性条件。通过OSS-bijection的概念，我们证明了闭包算子的一个子类与上述抽象复形家族之一之间的对应定理，这类似于闭包算子和摩尔系统之间的经典对应定理。接下来，我们展示了当集合系统属于上述家族之一时由集合系统引起的二元关系的扩展属性。最后，我们提供了在本工作中研究的三类抽象单纯复形之一的集合之间配对方面的表示结果。我们证明了一个闭包算子的子类与上述抽象复形家族之一之间的对应定理，这类似于闭包算子和摩尔系统之间的经典对应定理。接下来，我们展示了当集合系统属于上述家族之一时由集合系统引起的二元关系的扩展属性。最后，我们提供了在本工作中研究的三类抽象单纯复形之一的集合之间配对方面的表示结果。我们证明了一个闭包算子的子类与上述抽象复形家族之一之间的对应定理，这类似于闭包算子和摩尔系统之间的经典对应定理。接下来，我们展示了当集合系统属于上述家族之一时由集合系统引起的二元关系的扩展属性。最后，我们提供了在本工作中研究的三类抽象单纯复形之一的集合之间配对方面的表示结果。