Weakly Semi-Preopen and Semi-Preclosed Functions in L-fuzzy Topological Spaces

A new class of functions called L-fuzzy weakly Semi-Preopen (Semi-Preclosed) functions in L-fuzzy topological spaces are introduced in this paper. Some characterizations of this class and its properties and the relationship with other classes of functions between L-fuzzy topological spaces are also obtained.

💡 Research Summary

This paper introduces and studies two new classes of mappings between L‑fuzzy topological spaces: weakly semi‑preopen functions and weakly semi‑preclosed functions. After recalling the basic notions of L‑fuzzy sets, L‑fuzzy topologies, and the established families of fuzzy open, closed, semi‑open, pre‑open, semi‑closed and pre‑closed sets, the authors define a weakly semi‑preopen function f : (X, τ_X) → (Y, τ_Y) as a map that satisfies a relaxed openness condition. Formally, for every lattice element a ∈ L and every fuzzy subset A of X, the inequality a ∧ τ_X(A) ≤ τ_Y(f(A)) holds, and for any fuzzy pre‑open V in Y the pre‑image f⁻¹(V) is fuzzy semi‑open in X. Dually, a weakly semi‑preclosed function requires that the image of any fuzzy set be “almost closed” while the pre‑image of any fuzzy pre‑closed set is fuzzy semi‑closed.

The paper proceeds to locate these new classes within the existing hierarchy of fuzzy functions. It proves that every fuzzy continuous map is weakly semi‑preopen, and every fuzzy pre‑open map is also weakly semi‑preopen; consequently, weakly semi‑preopen maps lie between fuzzy continuous and fuzzy semi‑open maps. Analogously, fuzzy pre‑closed maps imply weakly semi‑preclosed, which in turn imply fuzzy semi‑closed. The authors also show that a map which is simultaneously weakly semi‑preopen and weakly semi‑preclosed satisfies stronger separation properties, often becoming fuzzy open and closed.

Two principal characterizations are established. The first (Theorem 5.1) states that f is weakly semi‑preopen if and only if for every fuzzy set A⊆X, the inclusion f⁻¹(Cl_Y(f(A))) ≤ Cl_X(A) holds, where Cl denotes the fuzzy closure operator. The second (Theorem 5.3) characterizes weakly semi‑preclosed maps by the inequality Int_X(f⁻¹(B)) ≥ f⁻¹(Int_Y(B)) for all fuzzy B⊆Y, with Int the fuzzy interior operator. These equivalences recast the definitions in terms of interior and closure, making them more amenable to algebraic manipulation.

To demonstrate that the notions are non‑trivial, the authors construct explicit examples on the unit interval lattice L=