Let K be a locally compact hyper group with left Haar measure and let L1 (K) be the complex Lebesgue space associated with it. Let L¥ (K) be the dual of L1 (K). The purpose of this paper is to present some necessary and sucient conditions for L¥ (K) * to have a topologically left invariant mean. Some Characterizations of amenable hyper groups are given.

Let R  be a Z-graded commutative ring with identity. Several Characterizations of graded distributive modules will be investigate.

In this paper, the concepts of L-concave structures, concave L-interior operators and concave L-neighborhood systems are introduced. It is shown that the category of L-concave spaces and the category of concave L-interior spaces are isomorphic, and they are both isomorphic to the category of concave L-neighborhood systems whenever L is a completely distributive lattice. Also, it is proved that these categories are all isomorphic to the category of L-convex spaces whenever L is a completely distributive lattice with an order-reversing involution operator.

Multivariate normal-Poisson model has been recently introduced as a special case of normal stable Tweedie models. The model is composed of a univariate Poisson variable, and the remaining variables given the Poisson one are independent Gaussian variables with variance the value of the Poisson component. Two Characterizations of this model are shown, first by variance function and then by generalized variance function which is the determinant of the variance function. The latter provides an explicit solution of a particular Monge-Ampère equation.

This article pays attention to the $\alpha$-migrativity of 2-uninorms with $\alpha\in [0,1]$ deeply. It describes the $\alpha$-migrativity of 2-uninorms completely, which generalizes and unifies some current existing Characterizations for the $\alpha$-migrativity of triangular norms, triangular conorms, uninorms, nullnorms, uni-nullnorms and null-uninorms, respectively.