Cycloidal algebras

Abstract

In this paper we introduce for an arbitrary algebra (groupoid, binary system) (X; *) a sequence of algebras (X; *) (n) = (X; a similar to), where x a similar to y = [x * y] (n) = x * [x * y] (n-1), [x * y](0) = y. For several classes of examples we study the cycloidal index (m, n) of (X; *), where (X; *) (m) = (X; *) (n) for m > n and m is minimal with this property. We show that (X; *) satisfies the left cancellation law, then if (X; *) (m) = (X; *) (n) , then also (X; *) (m-n) = (X; *)(0), the right zero semigroup. Finite algebras are shown to have cycloidal indices (as expected). B-algebras are considered in greater detail. For commutative rings R with identity, x * y = ax + by + c, a, b, c a a"e defines a linear product and for such linear products the commutativity condition [x * y] (n) = [y * x] (n) is observed to be related to the golden section, the classical one obtained for a"e, the real numbers, n = 2 and a = 1 as the coefficient b.