In this paper we generalize the notion of an @i-symmetric word, from an antimorphic involution, to an arbitrary involution @i as follows: a nonempty word w is said to be @i-symmetric if w=@a@b=@i(@b@a) for some words @a,@b. We propose the notion of @i-twin-roots (x,y) of an @i-symmetric word w. We prove the existence and uniqueness of the @i-twin-roots of an @i-symmetric word, and show that the left factor @a and right factor @b of any factorization of w as w=@a@b=@i(@b@a), can be expressed in terms of the @i-twin-roots of w. In addition, we show that for any involution @i, the catenation of the @i-twin-roots of w equals the primitive root of w. We also provide several characterizations of the @i-twin-rots of a word, for @i being a morphic or antimorphic involution.
['Lila Kari', 'Kalpana Mahalingam', 'Shinnosuke Seki']
Twin-roots of words and their properties