Dioid Partitions of Groups

TL;DR

This paper introduces dioid partitions of groups, explores their fundamental properties, and classifies specific partitions of cyclic groups of prime order, linking them to dioid structures.

Contribution

It defines dioid partitions of groups, establishes their properties, and classifies certain partitions of cyclic groups of prime order, extending the theory of Schur rings.

Findings

Dioid partitions induce dioid structures over groups.

Fundamental properties of dioid partitions are established.

Classification of three-part dioid partitions of cyclic groups of prime order.

Abstract

A partition of a group is a dioid partition if the following three conditions are met: The setwise product of any two parts is a union of parts, there is a part that multiplies as an identity element, and the inverse of a part is a part. This kind of a group partition was first introduced by Tamaschke in 1968. We show that a dioid partition defines a dioid structure over the group, analogously to the way a Schur ring over a group is defined. After proving fundamental properties of dioid partitions, we focus on three part dioid partitions of cyclic groups of prime order. We provide classification results for their isomorphism types as well as for the partitions themselves.