Introduction
In the realm of general rough sets, the relational approach explores various granular, pointwise, or abstract approximations and investigates rough objects [1-6]. These approximations can be extracted from information tables or deduced from data relevant to human or machine reasoning. A general approximation space consists of a pair where
represents a set and R is a binary relation (S and will be used interchangeably in this paper). These approximations of subsets of can be generated and studied at different levels of abstraction in theoretical rough set approaches. Beyond a general framework, the relational system becomes a crucial aspect due to the significance of approximations and related semantics. In many cases, S is interpreted as a set of attributes, and elements of S can be associated with a third element through reasoning, preference, or external decision-making mechanisms. This research aims to explore these scenarios from a minimalist standpoint, further enriching the knowledge concept in classical rough sets [7] and general rough sets [3, 8-10]. Thus, this study represents an extension of existing knowledge in the field.
Mereology, the study of parts and wholes, has been explored from various perspectives, including philosophical, logical, algebraic, topological, and applied approaches. In the field of mereology [9, 11, 12], it is argued that most binary part relations in human reasoning are at least antisymmetric and reflexive. The absence of transitivity in the parthood relation is primarily due to functional reasons that can lead to failures (see [11]), and to accommodate apparent parthood [12]. The study of mereology in the context of rough sets can be approached in at least two fundamentally different ways. In the approach adopted by the author [1, 2, 8, 10], the primary objective is to minimize data contamination by avoiding additional assumptions about the data in relation to the relevant semantic domain. On the other hand, numeric function-based approaches [13] define parthood based on the degree of rough inclusion or membership, which significantly differs from the former approach. Rough Y-systems and granular operator spaces, extensively introduced and studied by the author [1, 2, 8, 10, 12], are advanced abstract approaches in general rough sets that focus on the fundamental concepts of approximations, parthood, and granularity. The part-of relations mentioned in the previous paragraph can also be explored, and the relation R in a general approximation space can represent a parthood relationship. The author’s upcoming joint work investigates specific versions of parthood spaces, provides new results on parthood spaces, examines up-directedness in classical approximation spaces, and improves the formalism on granular operator spaces and variants. Additionally, potential applications of this research to education research contexts are outlined.
