김희식 2018-03-24T04:24:39Z 2018-03-24T04:24:39Z 2013-02 Mathematica Slovaca, Feb 2013, 63(1), P.33-40 0139-9918 1337-2211 http://www.degruyter.com/view/j/ms.2013.63.issue-1/s12175-012-0079-9/s12175-012-0079-9.xml http://hdl.handle.net/20.500.11754/51768 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 &gt; 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&quot;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&quot;e, the real numbers, n = 2 and a = 1 as the coefficient b. en Springer Science + Business Media cycloidal algebra B-algebra cycloidal index BCK-algebra linear product Cycloidal algebras Article 63 10.2478/s12175-012-0079-9 33-40 MATHEMATICA SLOVACA Han, Jeong Soon Kim, Hee Sik Neggers, J. 2009217949 S COLLEGE OF NATURAL SCIENCES[S] DEPARTMENT OF MATHEMATICS heekim