CHARACTRIZATION OF BI-GYROSEMIGROUPS: AN EMPHASIS ON ORDER 2

Abstract

The practical implementation and recognition of gyrogroups have been significantlyadvanced through their association with relativistically admissible velocities within special relativitytheory. This specific context provides a tangible representation where gyrogroup propertiescan be observed and studied effectively. By employing the Einstein velocity addition law todefine the binary operation within this space, researchers have successfully extended traditionalgroup theory concepts to encompass both gyrogroups and bi-gyrogroups. A bi-gyrogroup is agroup-Like Structure which satisfies the group condition, in addition to that, for any pair (a, b)in this structure, there are two automorphisms with this property that fulfill left and right associativityproperties. The study in this article is motivated by generalizing the bi-gyrogroupsand semigroups led to introducing bi-gyrosemigroups. Accordingly, in this paper, some classesof this structure are presented. Also, all bi-gyrosemigroups of order 2 are characterized.