Twin-roots of words and their properties

In this paper we generalize the notion of an ι-symmetric word, from an antimorphic involution, to an arbitrary involution ι as follows: a nonempty word w is said to be ι-symmetric if w = αβ = ι(βα) for some words α, β . We propose the notion of ιtwin-roots (x, y) of an ι-symmetric wordw. We prove the existence and uniqueness of the ι-twin-roots of an ι-symmetric word, and show that the left factor α and right factor β of any factorization of w as w = αβ = ι(βα), can be expressed in terms of the ι-twin-roots ofw. In addition, we show that for any involution ι, the catenation of the ι-twin-roots ofw equals the primitive root ofw. We also provide several characterizations of the ι-twin-rots of a word, for ι being a morphic or antimorphic involution. Crown Copyright© 2009 Published by Elsevier B.V. All rights reserved.