where (f + g)(u) = V_,,+,,,f(v) A f(w). Since E is countable, ZcB) can be defined for any countably complete Boolean algebra (ccBa) B where Z is the group of the integers. This kind of group was first (1962) studied by Balcerzyk [l]. However, it seems that not much attention was paid to such groups for a rather long period. Under the point of view in [l, Theorem 51 and [13, Proposition 11, it c...

In this article, we consider pair frames in Banach spaces and   introduce Banach pair frames. Some various concepts in the frame theory such as frames, Schauder frames, Banach frames and atomic decompositions are considered as   special kinds of (Banach) pair frames.  Some frame-like inequalities  for (Banach)  pair frames are presented. The elements that participant  in the construction of (Ba...

Let (G;N) be a pair of groups. In this article, first we con-struct a relative central extension for the pair (G;N) such that specialtypes of covering pair of (G;N) are homomorphic image of it. Second, weshow that every perfect pair admits at least one covering pair. Finally,among extending some properties of perfect groups to perfect pairs, wecharacterize covering pairs of a perfect pair (G;N)...