Quasi-MV algebras for complex fuzzy logic

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 $\alpha \oplus 0=\alpha$ 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 $S$. Next we construct a $\sqrt{\mathrm{\neg }}$ quasi-MV algebra over $S$ by adding an operation of square root of the negation. Moreover, implication connective and some derived connectives on $S$ are introduced. Furthermore, we construct a quasi-Wajsberg algebra over $S$ in which implication is a primitive connective. These algebraic structures are suitable for CFL.