1 = u for any word u; ua ш vb = (u ш vb)a + (ua ш v)b for any a,b ∈ A and any words u and v. The shuffle algebra on the alphabet A is defined to be the additive group R endowed with ш as multiplication. Radford's theorem now states that the Lyndon words are algebraically independent elements of this shuffle algebra, and generate it; thus, the shuffle algebra is isomorphic to a polynomial ring over k, with the indeterminates corresponding to the Lyndon words. See also Lexicographically minimal string rotation Morphic word Sturmian word Necklace (combinatorics) Notes References . . . . . . . . </ref> . . . . . . . Combinatorics on words