The purpose of this paper is to provide a detailed structure description of derived subpolygroups of polygroups. We introduce the concept of perfect and solvable polygroups and we give some results in this respect. Finally, we discuss on τ-multi-semi-direct hyperproduct of polygroups.

The aim of the article is to initiate a new study in the framework of algebraic hyperfields, with an applicative impact in cryptography. First we define the notions of generalized Weierstrass equation and elliptic hypercurve on Krasner hyperfields, as a generalization of the notion of elliptic curve on fields. Then we investigate properties of the hypergroups derived from elliptic hypercurves and of the associated H_v-groups. Finally, we present a class of canonical hypergroups, which can be used as an alphabet in a special cryptographic system.

In this article, we first introduce the notion of complementable semihypergroup, proving that the classes of simplifiable semigroups, groups, simplifiable semihypergroups, and complete hypergroups are examples of complementable semihypergroups. Then we define when two semihypergroups are disjoint and find examples of such semihypergroups. Finally, we discuss on the complementable property of K_H-hypergroups.

In this paper, we introduce generalized quadratic forms and hyperconics over quotient hyperfields as a generalization of the notion of conics on fields. Conic curves utilized in cryptosystems; in fact the public key cryptosystem is based on the digital signature schemes (DLP) in conic curve groups. We associate some hyperoperations to hyperconics and investigate their properties. At the end, a collection of canonical hypergroups connected to hyperconics is proposed.

In this paper, we study hypergroups determined by lattices introduced by Varlet and Comer, especially we enumerate Varlet and Comer hypergroups of orders less than 50 and 13, respectively.