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Natural sciences
- Other mathematical sciences and statistics not elsewhere classified
- Knowledge representation and reasoning
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Engineering and technology
- Other computer engineering, information technology and mathematical engineering not elsewhere classified
Grouping and ranking alternatives are two fundamental aspects of any decision-making process, ranging from everyday tasks to advanced scientific problems such as clustering, classification, and regression in machine learning. While grouping leads to a symmetric relation (indifference relation), ranking results in an antisymmetric relation (preference relation). If the additional relational property of transitivity is satisfied, grouping yields an equivalence relation, whereas ranking leads to a partial order relation. However, this transitivity property is far from trivial and is the subject of many paradoxes. For example, indifference/similarity is not necessarily transitive (e.g., the paradox of Poincaré, the paradox of Luce), and preference can lead to cycles (e.g., Rock-Paper-Scissors). These insights are primarily limited to classical Boolean binary relations, while my contributions have been centered on the context of fuzzy relations—a generalization of binary relations that allows expressing associations in a gradual manner. The main research lines, all related to the transitivity property, are as follows:
Reciprocal relations and stochastic orderings
A significant scenario arises when alternatives can be identified with random variables (e.g., yield, pollution). Stochastic orderings aim to rank these random variables. Various approaches are possible, such as one-versus-one and one-versus-all, with the notion of winning probabilities playing a central role within the relational framework of reciprocal relations. However, these methods often consider only pairwise dependencies, neglecting the entire dependency structure (which can be captured using the concept of an n-copula). Comparisons of random vectors remain largely unexplored, though there are likely connections with my work on additive preference structures.
Previous research revealed surprising results: the transitivity properties of reciprocal relations can be expressed as corresponding properties of fuzzy relations that are valid at the triplet level with a minimum probability threshold. These insights could pave the way for defining other mathematical properties in a similar frequentist manner, bridging overly rigid mathematical definitions with reality. Many research questions emerge from this perspective.
Ternary and higher-order relations
Most international research focuses on binary relations, i.e., pairwise relations. However, applications increasingly highlight the importance of ternary relations (e.g., in knowledge representation paradigms, higher-order interactions in ecological networks). Theoretical developments, however, lag behind. Recently, I completed an extensive study on compositions of ternary relations, which correspond to new transitivity properties. These still follow the classical thought pattern where restrictions on certain triplets arise from satisfying two clauses for related triplets. Such triplets may involve four or five points.
An unpublished result from Autumn 2024 unveiled a surprising new composition of ternary relations based on satisfying three clauses. The corresponding transitivity property plays a role in characterizing road systems, suggesting that research on the transitivity properties of ternary Boolean relations is still in its infancy—a state that also applies to the relational framework of ternary fuzzy relations. In this context, special attention will be given to revitalizing the concept of betweenness relations.
Multi-Point Properties
The properties of relations can be classified based on the number of points they constrain. For instance, transitivity of binary relations involves three points, whereas for ternary relations, it involves four or five points. Furthermore, there are properties of binary (fuzzy and reciprocal) relations that pertain to four points (e.g., the Ferrers property, which has already appeared in the context of winning probabilities). The goal is to develop a holistic approach to studying relational properties from the perspective of constraints on a given number of points.
Trellises
The absence of transitivity transforms a poset (i.e., a partially ordered set) into a psoset (i.e., a pseudo-ordered set), an object of limited interest. A crucial subclass of posets consists of lattices: posets where every two elements have a greatest lower bound (meet) and a least upper bound (join). Lattices play an essential role in many modern data analysis methods, such as formal concept analysis.
A recently rediscovered fascinating subclass of psosets is trellises, where the existence of meets and joins is preserved. The study of trellis-ordered semigroups, as a generalization of triangular norms (which play a crucial role in transitivity properties of fuzzy and reciprocal relations), is still in its infancy. The renewed interest in trellises raises numerous contemporary mathematical questions.
These seemingly parallel research lines form a coherent and mutually reinforcing whole, centered on the pivotal property of transitivity. This research is part of my personal work, collaborations with international researchers, and, occasionally, doctoral research projects. The insights gained subsequently inform more applied research in the Faculty of Bioscience Engineering, particularly in my research lines concerning knowledge-based, spatiotemporal, and predictive modeling—areas especially suitable for doctoral students.