Complex fuzzy logic (CFL) is an emerging topic of research in fuzzy logic. Due to the circle structured codomain of the complex membership function, algebraic structure (including MV-algebra) in traditional fuzzy logic cannot be easily transplanted to CFL. Quasi-MV algebras are almost identical to MV algebras, except $ \alpha\oplus\textbf{0} = \alpha $ does not always hold. In this paper, our goal is to derive some algebraic structures for CFL. We first construct a quasi-MV algebra in complex fuzzy logic by introducing negation and truncated sum in the unit disc of the complex plane $ \textbf{S} $. Next we construct a $ \sqrt{\neg} $ quasi-MV algebra over $ \textbf{S} $ by adding an operation of square root of the negation. Moreover, implication connective and some derived connectives on $ \textbf{S} $ are introduced. Furthermore, we construct a quasi-Wajsberg algebra over $ \textbf{S} $ in which implication is a primitive connective. These algebraic structures are suitable for CFL.
Citation: Songsong Dai. Quasi-MV algebras for complex fuzzy logic[J]. AIMS Mathematics, 2022, 7(1): 1416-1428. doi: 10.3934/math.2022083
Complex fuzzy logic (CFL) is an emerging topic of research in fuzzy logic. Due to the circle structured codomain of the complex membership function, algebraic structure (including MV-algebra) in traditional fuzzy logic cannot be easily transplanted to CFL. Quasi-MV algebras are almost identical to MV algebras, except $ \alpha\oplus\textbf{0} = \alpha $ does not always hold. In this paper, our goal is to derive some algebraic structures for CFL. We first construct a quasi-MV algebra in complex fuzzy logic by introducing negation and truncated sum in the unit disc of the complex plane $ \textbf{S} $. Next we construct a $ \sqrt{\neg} $ quasi-MV algebra over $ \textbf{S} $ by adding an operation of square root of the negation. Moreover, implication connective and some derived connectives on $ \textbf{S} $ are introduced. Furthermore, we construct a quasi-Wajsberg algebra over $ \textbf{S} $ in which implication is a primitive connective. These algebraic structures are suitable for CFL.
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Songsong Dai. Quasi-MV algebras for complex fuzzy logic[J]. AIMS Mathematics, 2022, 7(1): 1416-1428. doi: 10.3934/math.2022083
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