The family of rings the form
 \frac{\mathbb{Z}_{4}\left \langle x,y \right \rangle}{\left x^2-a,y^2-b,yx-xy-2(c+dx+ey+fxy) \rangle}
 is investigated which contains generalized Hamilton quaternions over $\Z_4$. These are local order 256. This has 256 contained in 88 distinct isomorphism classes. Of non-isomorphic rings, 10 minimal reversible nonsymmetric and 21 abelian reflexive nonsem...

Abstract Let G G be a finite nonabelian group. For any integer m ≥ 2 m\ge 2 , let A 1 , … , {A}_{1},\ldots ,{A}_{m} nonempty subsets of . If are mutually disjoint a...

Quaternion derivatives exist only for a very restricted class of analytic (regular) functions; however, in many applications, functions of interest are real-valued and hence not analytic, a typical case being the standard real mean square error objective function. The recent HR calculus is a step forward and provides a way to calculate derivatives and gradients of both analytic and non-analytic...

In this paper, we introduce and study the notion of precompacttopological generalized groups and some new results are given.