Compact symmetric triads and symmetric triads with multiplicities

TL;DR

This paper develops a classification theory for symmetric triads with multiplicities, focusing on their structure and applications to commutative compact symmetric triads, providing a comprehensive framework for understanding their properties.

Contribution

It introduces a classification of symmetric triads with multiplicities and characterizes those corresponding to commutative compact symmetric triads, expanding the theoretical understanding.

Findings

Classified abstract symmetric triads with multiplicities.

Identified symmetric triads with multiplicities related to commutative compact symmetric triads.

Provided two types of classifications based on equivalence relations.

Abstract

In this paper, we develop the theory of symmetric triads with multiplicities. First, we classify abstract symmetric triads with multiplicities. Second, we determine the symmetric triads with multiplicities corresponding to commutative compact symmetric triads. As applications, we give the classifications for commutative compact symmetric triads, which consist of two types depending on the choice of the equivalence relations.