This paper presents a new understanding of the multi-context handling of relational concept analysis (RCA). RCA can generate multiple concept lattices from data with circular dependencies, but the conventional RCA has the problem of returning only one set of concept lattices. To solve this problem, this paper defines a set of concept lattices that satisfy three conditions: 'well-formed', 'saturated', and 'self-supported' as 'admissible solutions'. We consider the RCA process from a functional perspective, and define the space of admissible solutions and the expansion and contraction functions that operate on the space. We prove that admissible solutions are common fixed points of these two functions, and show that the conventional RCA returns the minimum element of the set of admissible solutions. Furthermore, we construct an operation that generates a maximum element, and show that the set of admissible solutions is a complete sublattice of the interval between these two elements. We study in detail how this structure and the defined functions explore this structure.