The largest class of algebraic hyper structures satisfying the module like axioms is the Hv-module. In this paper, we consider the category of Hv-modules and prove that the direct limit always exists in this category. Direct limits are defined by a universal property, and so are unique. The most powerful tool in order to obtain a module from a given Hv-module is the quotient out procedure. To use this method we consider the fundamental equivalence relation ε * , and then prove some of the results about the connection between the fundamental modules, direct systems and direct limits.

In this paper, we will present the formulation of the direct limit of a direct system of fuzzy multialgebras and some  fundamental properties of this formulation. Moreover, we will show that the direct limit of a direct system of multialgebras  is isomorphic to the direct limit of any direct system whose carrier is cofinal with the carrier of the given system.

Leedham-Green and McKay [ Acta Math.137( 1967) 99-150] introduced the generalized version of the Baer-invariant of a group with respect to two varieties of groups. A group G is called capable if there exists a group H such that G=H/Z(H). In this paper , we generalize some properties of capability of direct product of groups with respect to two varieties of groups and direct Limits.Moreover, we survey some properties of the Baer - invariant of a pair of groups with respect to two varieties of groups and direct Limits.

In this paper we introduce the concept of generalized Baer-invariant of a pair of groups with respect to two varieties ν and w of groups. We give some inequalities for the generalized Baer-invariant of a pair of finite groups, when ν is considered to be the Schur-Baer variety. Further, we present a sufficient condition under which the order of the generalized Baer-invariant of a pair of finite groups divides the order of the generalized Baer-invariant of a pair of their factor groups. Moreover, we prove that the generalized Baer-invariant of a pair of groups commutes with direct limit.

In this paper, we introduce the concept of left(right)-G sets by external hyperoperations and some examples presented. We con-struct quotient left(right)-G sets by regular (strongly) regular relations. Also, we consider fundamental relation as a smallest strongly regular relations and by complete parts concepts introduce an equivalence re-lation that is coincide with fundamental relation. The main purpose of this paper is to introduce the concepts of tensor product and direct limit on G-sets of n-ary semihypergroups that are non-additive modi, cation of classical construction in module theory. This concept is crucially important in homological algebra and several properties are found and examples are presented.