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Multiple-valued majority functions were defined in 1980 as a group which differ from; threshold functions, but the majority functions only consist of a subset of Kleenean functions. This paper defines an extension of the ternary majority function using cyclic operation to I/O values of the function. The extended ternary majority functions are functionally complete on a ternary logic system. We apply this function to synsthesize evolvable logic system. The goal of the discussion in the latter half is to devise a method of representing any ternary logic function using the extended ternary majority functions. GA is used as a tool, and an advanced method applicable to many variable case is proposed together with some experiments.