r-fuzzy \(\alpha\)-open and r-fuzzy pre-open sets in fuzzy bitopological spaces (Q2706961)

Studying properties of \([0,1]\)-bitopological spaces (or fuzzy bitopological spaces in the sense of Chang) \textit{S. Sampath Kumar} [Soochow J. Math. 21, No. 3, 263-272 (1995; Zbl 0854.54031); Fuzzy Sets Syst. 64, No 3, 421-426 (1994; Zbl 0843.54015)]), introduced notions of \(\alpha\)-open, pre-open and semi-open fuzzy sets in a \([0,1]\)-bitopological space and the properties of pairwise continuity, pairwise \(\alpha\)-continuity and pairwise semi-continuity of mappings of \([0,1]\)-bitopological spaces. In this paper these notions are ``levelwise'' extended to the case of \(([0,1]\)-)fuzzy bitopological spaces \((X,\tau_1,\tau_2)\) (where \(\tau_1,\tau_2:[0,1]^X\to[0,1]\) are \([0,1]\)-fuzzy topologies in the sense of the reviewer [\textit{A. P. Šostak}, Rend. Circ. Mat. Palermo, II. Ser. Suppl. 11, 89-103 (1985; Zbl 0638.54007)], thus coming to such concepts as an \(r\)-\(\alpha\)-open fuzzy set, an \(r\)-pre-open fuzzy set, a fuzzy pairwise continuous and fuzzy pairwise \(\alpha\)-continuous function, fuzzy pairwise open function, etc. Basic properties of the introduced notions are studied. In particular, the paper contains some examples illustrating relations between them.