Relational concept analysis (RCA) is an extension of formal concept analysis allowing to deal with several related contexts simultaneously. It has been designed for learning description logic theories from data and used within various applications. A puzzling observation about RCA is that it returns a single family of concept lattices although, when the data feature circular dependencies, other solutions may be considered acceptable. The semantics of RCA, provided in an operational way, does not shed light on this issue. In this report, we define these acceptable solutions as those families of concept lattices which belong to the space determined by the initial contexts (well-formed), cannot scale new attributes (saturated), and refer only to concepts of the family (self-supported). We adopt a functional view on the RCA process by defining the space of well-formed solutions and two functions on that space: one expansive and the other contractive. We show that the acceptable solutions are the common fixed points of both functions. This is achieved step-by-step by starting from a minimal version of RCA that considers only one single context defined on a space of contexts and a space of lattices. These spaces are then joined into a single space of context-lattice pairs, which is further extended to a space of indexed families of context-lattice pairs representing the objects manippulated by RCA. We show that RCA returns the least element of the set of acceptable solutions. In addition, it is possible to build dually an operation that generates its greatest element. The set of acceptable solutions is a complete sublattice of the interval between these two elements. Its structure and how the defined functions traverse it are studied in detail.

翻译：关系概念分析（RCA）是形式概念分析的扩展，允许同时处理多个关联上下文。它被设计用于从数据中学习描述逻辑理论，并已在多种应用中得以应用。关于RCA的一个令人费解的观察是，尽管当数据存在循环依赖时其他解也可能被视为可接受，但该算法仅返回单一的概念格族。RCA的语义以操作化方式给出，未能阐明这一问题。在本报告中，我们将这些可接受的解定义为满足以下条件的概念格族：属于初始上下文确定的空间（良构）、无法缩放新属性（饱和）、且仅引用该族中的概念（自支撑）。我们通过定义良构解空间及其上的两个函数（一个扩张函数，一个压缩函数）来采用功能化视角审视RCA过程。研究表明，可接受的解正是这两个函数的公共不动点。这一结论通过逐步构建实现：首先从仅考虑单一上下文（定义在上下文空间和格空间上）的RCA最小版本出发，然后将这两个空间合并为单个上下文-格对空间，进一步扩展为表示RCA操作对象的索引化上下文-格对族空间。我们证明RCA返回可接受解集的最小元。此外，可以对偶地构造生成其最大元的操作。可接受解集是位于这两个元素之间的区间的完全子格，其结构及定义函数在该子格上的遍历方式均得到详细研究。