The Dual of Generalized Fuzzy Subspaces

In fuzzy algebra, fuzzy subspaces are basic concepts. They had been introduced by Katsaras and Liu 1 in 1977 as a generalization of the usual notion of vector spaces. Since then, many results of fuzzy subspaces had been obtained in the literature 1–4 . Moreover, many researches in fuzzy algebra are closely related to fuzzy subspaces, such as fuzzy subalgebras of an associative algebra 5 , fuzzy Lie ideals of a Lie algebra 6 , fuzzy subcoalgebras of a coalgebra 7 . Hence fuzzy subspaces play an important role in fuzzy algebra. In 1996, Abdukhalikov 8 defined the dual of fuzzy subspaces as a generalization of the dual of kvector spaces. This notion was also studied and applied in many branches 2, 7–9 , especially in the fuzzy subcoalgebras 7 and fuzzy bialgebras 9 . After the introduction of fuzzy sets by Zadeh 10 , there are a number of generalizations of this fundamental concept. So it is natural to study algebraic structures connecting with them. In this paper, we aim our attention at the dual of vector space in intuitionistic fuzzy sets, interval-valued fuzzy sets, and interval-valued intuitionistic fuzzy sets for our further researches.