Variable-basis fuzzy interior operators

For a topological space it is well-known that the associated closure and interior operators provide equivalent descriptions of set-theoretic topology; but it is not generally true in other categories, consequently it makes sense to define and study the notion of interior operators I in the context of fuzzy set theory, where we can find categories in a lattice-theoretical context. Fuzzy interior operators have been studied by U. Hohle, A. Sostak and others, (1999), these works were used to describe L-topologies on a set X. More recently, M. Diker, S. Dost and A. Ugur (2009) present interior and closure operators on texture spaces in the sense of Cech, and F. G. Shi(2009) studies interior operators via L-fuzzy neighborhood systems. The aim of this paper is to propose a more general theory of variable basis fuzzy interior operators, employing both categorical tools and the lattice theoretical foundations investigated by S. E. Rodabaugh (1999), where tha lattices are usually non-complemented. furthermore, we construct some topological categories.

💡 Research Summary

The paper addresses a fundamental asymmetry that arises when one attempts to translate the classical equivalence between closure and interior operators from ordinary topological spaces to more general categorical settings, especially those involving fuzzy sets valued in a lattice L. In classical topology, the interior operator I and the closure operator C are mutually definable via complementation, but this duality breaks down when the underlying lattice is non‑complemented or when the “basis” of the fuzzy structure varies. To overcome this limitation, the authors introduce the notion of a variable‑basis fuzzy interior operator.

The central idea is to allow the lattice L that supplies truth‑values to change from one context to another, rather than fixing a single L for the whole theory. For a set X and a lattice L, an L‑valued fuzzy subset is a map f : X → L, and the collection of all such maps is denoted L^X. An interior operator I_X : L^X → L^X is required to satisfy three axioms: (i) expansivity (I_X(f) ≥ f), (ii) monotonicity (f ≤ g ⇒ I_X(f) ≤ I_X(g)), and (iii) normality (I_X(⊤) = ⊤). Crucially, the operator depends on the chosen lattice L; if a different lattice M is used for the same underlying set, a generally different interior operator I_X^M results. This flexibility captures a wide variety of fuzzy topological structures that were previously treated only as special cases.

To place these operators within a categorical framework, the authors define objects as triples (X, L, I_X) and morphisms as pairs (φ, ψ) where φ : X → Y is a set map and ψ : L → M is a lattice homomorphism. The pair must satisfy a compatibility condition with the interior operators: ψ ∘ I_X = I_Y ∘ ψ ∘ φ_, where φ_ is the induced map L^X → M^Y. This condition ensures that the interior structure is preserved under the morphism. The resulting category, denoted VIF (Variable‑basis Interior Fuzzy), generalizes the classical category of topological spaces (Top) and the L‑fuzzy topological category (L‑Top).

In parallel, the authors construct a dual category VCF (Variable‑basis Closure Fuzzy) by defining closure operators C_X : L^X → L^X that satisfy extensivity, monotonicity, and minimality (C_X(⊥) = ⊥). Because the lattice may be non‑complemented, C_X and I_X are not automatically complements of each other. Nevertheless, the paper proves that VIF and VCF are di‑adjoint: each interior operator is the left adjoint of a corresponding closure operator and vice‑versa. This adjunction mirrors the classic interior‑closure adjunction in ordinary topology but holds in the much broader setting of variable bases and non‑complemented lattices.

The authors then demonstrate that several earlier works appear as special instances of their framework. When L is a complete Boolean algebra, VIF collapses to the Höhle‑Sostak L‑topology category. The interior operators studied by Diker, Dost, and Ugur on Cech‑type texture spaces correspond to VIF objects where the lattice is fixed but the underlying texture structure varies. Shi’s L‑fuzzy neighborhood systems are recovered when the lattice homomorphism ψ is the identity. Thus, the variable‑basis approach unifies disparate strands of fuzzy topology under a single categorical umbrella.

Finally, the paper outlines future directions. The flexibility of varying the basis lattice suggests applications to multi‑layered fuzzy systems, dynamic topologies where the truth‑value lattice evolves over time, and connections with homological algebra via the di‑adjunction. Moreover, the construction of VIF and VCF opens the door to developing sheaf‑theoretic and cohomological tools for fuzzy spaces, as well as exploring logical interpretations in many‑valued logics. In summary, the work provides a robust, category‑theoretic foundation for fuzzy interior operators that accommodates non‑complemented lattices and variable bases, thereby extending and unifying existing fuzzy topological theories.