The Definitions and Some Elementary Properties of Polygroup

Abstract

The polygroup concept has found applications in many branches of mathematics such as analysis, algebra, geometry and fuzzy sets. The notion of quasi-canonical hypergroup called polygroup, which is a generalization of the notion of a group. A hyper-structure admits more than one output under the operation, which is called a hyper-operation. We generalize the notion of polygroup structure induced by the double cosets, which is also a generalization of the group object. There are only a few studies that have developed polygroup under hyper-operation. In our previous work, some of its properties such as normal subpolygroups, maximal subpolygroups and chain conditions were proved. In this paper, we construct a new polygroup from a given polygroup. Also, we show that if G, KandH are polygroups, then G∕∕H∕∕K∕∕H≅G∕∕K which is induced by the double cosets of a group.