Let (R, m) be a Noetherian local ring. Two notions of filter REGULAR sequence and generalized local cohomology module with respect to a PAIR of ideals are introduced, and their properties are studied. Some vanishing and non-vanishing theorems are given for this generalized version of generalized local cohomology module.

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 (Banach) PAIR frames are characterized. It is shown that a Banach space X has a Banach PAIR frame with respect to a Banach scalar sequence space `, when it is precisely isomorphic to a complemented subspace of `. It is shown that if we are allowed to choose the scalar sequence space, PAIR frames and Banach PAIR frames with respect to the chosen scalar sequence space denote the same concept.

Hypergroups that have at least one identity element and where each element has at least one inverse are called REGULAR hypergroup. In this regards, for a REGULAR hypergroup $H$, it is shown that there exists a correspondence between the set of all strongly REGULAR relations on $H$ and the set of all normal subhypergroups of $H$ containing $S_{\beta}$. More precisely, it has been proven that for every strongly REGULAR relation $\rho$ on $H$, there exists a unique normal subhypergroup of $H$ containing $S_{\beta}$, such that its quotient is a group, isomorphic to $H/\rho$. Furthermore, this correspondence is extended to a lattice isomorphism between them.

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In this paper we investigate the PAIR difference cordial labelingbehaviour of diamond ladder graph,lattitude ladder, octopus graph,pagodagraph, planter graph and semi jahangir graph. Prime labeling behaviour of plantergraph, duplication of planter graph, fusion of planter graph,switching of plantergraph, joining of two copies of planter graph were studied by A.Edward samueland S.Kalaivani[4]. Dafik, Riniatul Nur Wahidah hve been examined the RainbowAntimagic coloring of special graphs like volcano, sandat graph, sunflower,octpus,semijahangir [3]. Prihandini,R M., at.el have been studied the elegant labeling ofshackle graphs and diamond ladder graphs [16]. Classical meanness of some graphssuch as one-side step graph,double-sided step graph,grid,slanding ladder,diamondladder,lattitude ladder was studied by Alanazi et. al [1]. In [17] Yeni Susanti et.al studied the edge odd geaceful labeling behaviour of prism, antiprism, cartesianproduct graphs. The notion of PAIR diference cordial labeling of a graph was introduced in [7].

In this paper, we introduce nilpotent and solvable PAIR of polygroups. By a PAIR of polygroups (P, N), we mean a polygroup P with a normal sub-polygroup N of P. In addition, we obtain a necessary and su cient condition between nilpotency (solvability) of PAIR of polygroups and PAIR of groups. In particular, we extend the theory of groups to a PAIR of polygroups. Finally, we study the relationship between nilpotent and solvable PAIR of poly-groups.