On Singh’s dressed epsilon perspective of multigroup

Abstract

This paper investigates some aspects of multigroups from Singh's perspective. It introduces a structured approach to analyzing the aspects of multigroups presented in this paper using dressed epsilon notation. We begin by defining the hierarchical decomposition of multisets, establishing that each -level reference set in the hierarchical decomposition of a multiset over a group is itself a subgroup. We then present a fundamental characterization of multigroups by proving that a multiset is a multigroup over a set  if and only if . Additionally, we define the sets and prove that both are subgroups of  using Singh’s dressed epsilon notation. Our work further investigates the algebraic properties of multigroupsand establishes criteria for commutativity. We also demonstrate that while the intersection of two multigroups is always a multigroup, their union does not necessarily inherit this structure. The concept of submultigroup is introduced to formalize the relationship between two multigroups. Finally, we establish the equivalence between certain multigroup properties, such as the symmetry of multisets based on product of elements and conjugate conditions.