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。它旨在从数据中学习描述逻辑理论，并已应用于多种场景。关于RCA的一个令人困惑的观察是，它只返回单一的概念格族，尽管当数据具有循环依赖时，其他解也可能被认为可接受。RCA的语义以操作方式给出，并未阐明此问题。在本报告中，我们将这些可接受解定义为那些属于初始上下文确定的空间（良构的）、不能缩放新属性（饱和的）、且仅引用族内概念（自支撑的）的概念格族。我们采用功能视角看待RCA过程，通过定义良构解空间以及该空间上的两个函数（一个扩张函数、一个收缩函数），证明可接受解是这两个函数的公共不动点。这一结论是通过逐步方式实现的：首先从仅考虑单一上下文的RCA最小版本开始，该版本定义在上下文空间和格空间之上。随后将这些空间合并为单一的上下文-格对空间，并进一步扩展为表示RCA所操作对象的索引化上下文-格对族空间。我们证明RCA返回的是可接受解集合的最小元。此外，可以对偶地构造生成其最大元的操作。可接受解集合是这两个元素区间内的完备子格。本文详细研究了该集合的结构以及所定义函数在其上的遍历方式。