Operation Insertion on the Conjugacy and Commutativity of Words

In this paper, we do a theoretical study of some of the notions in combinatorics of words with respect to the insertion operation on words. We generalize the classical notions of conjugacy and commutativity of words to insertion-conjugacy and insertion-commutativity of words. We define and study properties of such words (i.e.) of words u, v and w such that (formula presented).