Ambiguous representations as fuzzy relations between sets

Crisp and $L$-fuzzy ambiguous representations of closed subsets of one space by closed subsets of another space are introduced. It is shown that, for each pair of compact Hausdorff spaces, the set of (crisp or $L$-fuzzy) ambiguous representations is a lattice and a compact Hausdorff Lawson upper semilattice. The categories of ambiguous and $L$-ambiguous representations are defined and investigated.

💡 Research Summary

The paper introduces a novel mathematical framework called “ambiguous representations” to model the uncertain correspondence between closed subsets of two compact Hausdorff spaces. Starting from basic notions of binary and ternary relations, the authors treat a relation as a closed subset of the product space, and then extend this to L‑fuzzy sets and relations where L is a complete distributive lattice equipped with a continuous, associative, commutative, and infinitely distributive binary operation . An L‑fuzzy relation R : X × Y → L can be represented by its subgraph sub R = { (x, y, α) | α ≤ R(x, y) } and its composition with another fuzzy relation S is defined by (R ⊚ S)(x, z) = sup_{y∈Y} R(x, y) * S(y, z).

The central definition (Definition 2.3) of a crisp ambiguous representation is a subset R ⊂ exp X × exp Y satisfying three conditions: (a) monotonicity with respect to inclusion (if A′⊂A and B⊂B′ and (A, B)∈R then (A′, B′)∈R), (b) totality (for every A∈exp X, (A, Y)∈R), and (c) closedness of the “image” AR = { B | (A, B)∈R } in exp Y. Intuitively, A can be “represented” by any B that belongs to AR, capturing the idea that a set of objects in one space may be approximated or identified by a set of objects in another space under uncertainty.

The authors then lift this notion to the L‑fuzzy setting. An L‑valued capacity c on a compactum X is a monotone map c: exp X ∪ {∅} → L satisfying c(∅)=0, c(X)=1, and an upper semicontinuity condition expressed via neighborhoods in L. The subgraph sub c = { (F, α) | α ≤ c(F) } is a closed subset of exp X × L, and the family of all such capacities M_L X is equipped with a compact Hausdorff topology generated by basic open sets O⁺(U, V) and O⁻(F, V). Lemma 1.2 characterizes when a subset of exp X × L is the subgraph of a capacity, providing the foundation for defining L‑fuzzy ambiguous representations as closed subsets of exp X × exp Y × L satisfying analogous monotonicity and closure conditions.

A major structural result is that the collection of all (crisp or L‑fuzzy) ambiguous representations between fixed compacta X and Y forms a lattice under pointwise intersection (meet) and a suitably defined closure of union (join). Moreover, this lattice is a compact Hausdorff Lawson upper semilattice: each point possesses a local base consisting of upper sub‑semilattices, and the order topology coincides with the Vietoris topology on the hyperspace of closed subsets. The Lawson property guarantees the existence of least upper bounds and greatest lower bounds that vary continuously with respect to the underlying spaces.

From a categorical perspective, the authors define two categories: Amb, whose objects are compact Hausdorff spaces and morphisms are crisp ambiguous representations, and L‑Amb, where morphisms are L‑fuzzy ambiguous representations. Composition of morphisms is given by the relational composition described earlier, and the authors verify that composition preserves the required monotonicity, totality, and closedness, thus making these structures genuine categories. They also discuss conditions under which a morphism admits an “inverse” ambiguous representation, providing a notion of reversible approximation between spaces.

The paper concludes with illustrative examples and potential applications. One example models the relationship between a set of test scores (X) and a set of students (Y), where a subset of scores A is “likely” to be the score set of a subset of students B; the ambiguous representation captures this probabilistic association. Another motivating scenario involves image recognition: a digitized photograph of a character may be shifted slightly, leading to many possible pixel configurations that are all acceptable approximations of the original character. By treating the original character and its shifted versions as closed subsets in appropriate function spaces, ambiguous representations can formalize the tolerance to small, structured perturbations versus random noise.

Overall, the paper provides a rigorous topological and algebraic foundation for modeling uncertain set-to-set correspondences, extending classical fuzzy set theory and rough set ideas. By integrating hyperspace topology, capacity theory, and Lawson lattice theory, it opens new avenues for handling imprecise data in areas such as image analysis, decision making, and information aggregation, where relationships between collections rather than individual elements are central.