Powers and Alternative Laws

A groupoid is alternative if it satisfies the alternative laws x(xy) = (xx)y and x(yy) = (xy)y. These laws induce four partial maps on N+ × N+ (r, s) 7→ (2r, s− r), (r − s, 2s), (r/2, s + r/2), (r + s/2, s/2), that taken together form a dynamical system. We describe the orbits of this dynamical system, which allows us to show that nth powers in a free alternative groupoid on one generator are well-defined if and only if n ≤ 5. We then discuss some number theoretical properties of the orbits, and the existence of alternative loops without two-sided inverses. 1. Alternative laws and the induced dynamical systems Let G be a free groupoid with one generator x. The elements of G are (correctly parenthesized) words built from the single letter x. The length |w| of a word w is the number of letters in w. For a positive integer n we denote by x any of the words of length n in G. Note that there are precisely cn such words, where cn is the nth Catalan number defined by the recursive relation c0 = 1, c1 = 1, cn+1 = c1cn + c2cn−1 + · · ·+ cn−1c2 + cnc1, cf. [7]. A groupoid is said to be left alternative if it satisfies the left alternative law x(xy) = (xx)y. Dually, it is right alternative if it satisfies the right alternative law x(yy) = (xy)y. A groupoid that is both left alternative and right alternative is called alternative. (When dealing with algebras, the flexible law x(yx) = (xy)x is counted among alternative laws, and hence alternative algebras by definition satisfy the flexible law in addition to the left and right alternative laws. Our terminology is common for nonassociative structures with one binary operation.) Let A be the free alternative groupoid with generator x. Then A consists of equivalence classes of G, where two elements of G are equivalent if and only if they can be obtained from each other by finitely many applications of the alternative laws. For instance, the equivalence class of (xx)(xx) consists of all possible powers x, as is immediately seen from x((xx)x) = x(x(xx)) = (xx)(xx) = ((xx)x)x = (x(xx))x and from the fact that c4 = 5. Thus the words of the form x form an equivalence class in A, i.e., x is well-defined. The goal of this paper is to determine for which n > 0 the power x is welldefined in A, and to investigate related questions. In Sections 5 and 6 we turn our attention to alternative loops without two-sided inverses. The following concept proves useful in all of these tasks: 1991 Mathematics Subject Classification. Primary: 20N02, Secondary: 20N05, 37E99.