The notion of cubic set was introduced in [ 23 ] as a generalization of the notion of fuzzy sets and intuitionistic fuzzy sets. In this paper, we initiate a study of cubic sets in left almost semihypergroups. By using the concept of cubic sets, we introduce the notion of cubic sub LA-semihypergroups (hyperideals and bi-hyperideals) and discuss some basic results on cubic sets in LA-semihypergroups.

Using a new definition, with respect to [21], for normal hyperideals in Krasner ( m, n )-hyperrings, we show that the corresponding quotient structures are ( m, n )-rings. We prove equivalency of (strongly) regular and (strongly) compatible relations on n -ary hypergroups. Also, the connection between ( m, n )-hyperfields and maximal normal hyperideals is investigated.

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.

An ordered Krasner hyperring is a Krasner hyperring (R , + , ·) besides a partial order relation ≤ that satisfies the monotone conditions. In this paper, we introduce the concept of fuzzy hyperideals, fuzzy quasi-hyperideals and fuzzy bi-hyperideals

This paper presents a series of important results from the theory of n-hypergroups. Connections with binary relations and with lattices are presented. Special attention is paid to the fundamental relation and to the commutative fundamental relation. In particular, join n-spaces are analyzed.