Equations in polyadic groups

TL;DR

This paper investigates the solution sets of equations in polyadic groups, establishing a key equivalence with ordinary groups regarding the property of being equational noetherian, and characterizing their algebraic structures.

Contribution

It proves that a polyadic group is equational noetherian if and only if its underlying group is, and describes the structure of coordinate polyadic groups in this context.

Findings

Polyadic groups are equational noetherian iff their underlying groups are.

Structure of coordinate polyadic groups in equational noetherian polyadic groups is characterized.

Provides a bridge between properties of polyadic groups and their underlying groups.

Abstract

Systems of equations and their solution sets are studied in polyadic groups. We prove that a polyadic group $(G, f)=\mathrm{der}_{\theta, b}(G, \cdot)$ is equational noetherian, if and only if the ordinary group $(G, \cdot)$ is equational noetherian. The structure of coordinate polyadic group of algebraic sets in equational noetherian polyadic groups are also determined.