QUASIGROUP STRING PROCESSING: PART 4

A b s t r a c t: Given a ¯nite alphabet A and a quasigroup operation ¤ on the set A, in earlier paper of ours we have de¯ned the quasigroup transformation E : A+ ! A+, where A+ is the set of all ¯nite strings with letters from A. Here we present several generalizations of the transformation E and we consider the conditions under which the transformed strings have uniform distributions of n-tuples of letters of A. The obtained results can be applied in cryptography, coding theory, de¯ning and improving pseudo random generators, and so on.

An Affine Regular Icosahedron Inscribed in an Affine Regular Octahedron in a GS-Quasigroup

A golden section quasigroup or shortly a GS-quasigroup is an idempotent quasigroup which satises the identities a\dot (ab \dot c) \dot c = b; a\dot (a \dot bc) \dot c = b. The concept of a GS-quasigroup was introduced by VOLENEC. A number of geometric concepts can be introduced in a general GS-quasigroup by means of the binary quasigroup operation. In this paper, it is proved that for any affine regular octahedron there is an affine regular icosahedron which is inscribed in the given affine regular octahedron. This is proved by means of the identities and relations which are valid in a general GS-quasigrup. The geometrical presentation in the GS-quasigroup C(\frac{1}{2} (1 +\sqrt{5})) suggests how a geometrical consequence may be derived from the statements proven in a purely algebraic manner.

The Generalized Equations of Bisymmetry Associativity and Transitivity on Quasigroups

The generalized equations of bisymmetry, associativity and transitivity are, respectively,(1)(x1y)2(z3u) = (x4z)5(y6u)(2)(x1y)2z = x3(y4z)(3)(x1z)2(y3z) = x4y.The numbers 1, 2, 3,…, 6 represent binary operations and x, y, z and u are taken freely from certain sets.We shall be concerned with the cases in which x, y, z, and u are from the same set and each operation is a quasigroup operation. Under these conditions the solution of all three equations is known [1], [2]; equations (1) and (3) having been reduced to the form of (2) and a solution of (2) being given. We wish to present a new approach to these equations which we feel has the advantages that the equations may be resolved independently, the motivation behind the proof is clear, and the method lends itself to application on algebraic structures weaker than quasigroups. (Details of these generalizations will be given elsewhere.)