Cycloidal algebras

In this paper we introduce for an arbitrary algebra (groupoid, binary system) (X; *) a sequence of algebras (X; *) _n = (X; °), where x ° 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 ∈ ? 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 ?, the real numbers, n = 2 and a = 1 as the coefficient b.