On substructures and root sets in antimultigroups and their direct products

Abstract

This paper presents an extension of the direct product operation to antimultigroups. We prove that the direct product of two antimultigroups is itself an antimultigroup, preserving the defining axioms under Cartesian pairing. We introduce and analyze the main substructures of antimultigroups. These substructures include strong and weak upper and lower cuts, and show that each type of cut forms a sub-antimultigroup. Also, we examine the behavior of root sets and the structural connections between cuts under union and intersection. This leads to the establishment that such operations yield sub-antimultigroups under suitable conditions. These findings contribute to a deeper understanding of the structure of antimultigroups. Thus, it lays the groundwork for further developments in antimultigroup theory.