Intuitionistic H-fuzzy relations

In 1965, Zadeh [30] introduced a concept of a fuzzy set as the generalization of a crisp set. Also, in 1971, he introduced a fuzzy relation naturally, as a generalization of a crisp relation in [31]. Nel [27] introduced the notion of a topological universe which implies concrete quasitopos [1]. Every topological universe satisfies all the properties of a topos except one condition on the subobject classifier. The notion of a topological universe has already been put to effective use in several areas of mathematics in [24, 25, 28]. In 1980, Cerruti [8] introduced the category of L-fuzzy relations and investigated some of its properties. After that time, Hur [14] introduced the category Rel(H) of the fuzzy relational spaces with a complete Heyting algebra H as a codomain and he studied the category Rel(H) in the sense of a topological universe. In 1983, Atanassov [2] introduced the concept of an intuitionistic fuzzy set as the generalization of fuzzy sets and he also investigated many properties of intuitionistic fuzzy sets (cf. [3]). After that time, Banerjee and Basnet [4], Biswas [6], and Hur and his colleagues [15, 16, 17, 20] applied the concept of intuitionistic fuzzy sets to algebra. Also, Çoker [9], Hur and his colleagues [21], and S. J. Lee and E. P. Lee [26] applied one to topology. In particular, Hur and his colleagues [18] applied the notion of intuitionistic fuzzy sets to topological group. In this paper, we introduce the category IRel(H) of intuitionistic H-fuzzy relational spaces and study the category IRel(H) in a topological universe viewpoint. In particular, we show that IRel(H) satisfies all the conditions of a topological universe over Set except