Simplicial complexes (here briefly complexes) are set systems on an arbitrary set which are object of study in many areas of both mathematics and theoretical computer science. Usually, they are investigated over finite sets. However, in general, when we consider an arbitrary set Ω (not necessarily finite) and a complex 𝒞 on Ω, the most natural property related to finiteness is the following: for any subset X of Ω, if F ∈𝒞 for all finite subsets F of X, then X ∈𝒞. We call locally finite any complex 𝒞 having such a property. Bearing in mind some motivations and constructions derived from the analysis of information systems in rough set theory, in this paper we associate with any locally finite complex 𝒞 a corresponding pre-closure operator σ𝒞 and, through it, we establish several properties of 𝒞. Next, we investigate the main features of the specific sub-class of locally finite complexes 𝒞 for which σ𝒞 is a closure operator. We call these complexes closable and exhibit a particular family of closable locally finite complexes using left-modules on rings with identity. Finally, we establish a representation result according to which we can associate a pairing structure with any closable locally finite complex.

中文翻译：

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局部有限复合体、模块和广义信息系统

单纯复形（这里简要配合物) 是任意集合上的集合系统，它们是数学和理论计算机科学的许多领域的研究对象。通常，它们是在有限集上研究的。然而，一般来说，当我们考虑任意集合时Ω（不一定是有限的）和一个复杂的𝒞在Ω，与有限性相关的最自然的性质如下：对于任何子集X的Ω， 如果F ∈𝒞对于所有有限子集F的X， 然后X ∈𝒞. 我们称之为局部有限任何复杂的𝒞拥有这样的财产。考虑到从粗糙集理论中信息系统分析得出的一些动机和结构，在本文中，我们将与任何局部有限复形联系起来𝒞相应的关闭前操作员σ𝒞并且，通过它，我们建立了几个属性𝒞. 接下来，我们研究局部有限复形的特定子类的主要特征𝒞为此σ𝒞是一个闭包运算符。我们称这些复合物可关闭的并使用具有恒等式的环上的左模展示一个特定的可闭局部有限复形族。最后，我们建立了一个表示结果，据此我们可以将配对结构与任何可闭合的局部有限复形联系起来。