Classification of involutive commutative two-valued groups

TL;DR

This paper provides a complete classification of finitely generated involutive commutative two-valued groups, introducing three series and establishing isomorphism conditions, with extensions to topological and algebraic cases.

Contribution

It introduces a comprehensive classification framework for involutive commutative two-valued groups, including principal, unipotent, and special series, and extends results to topological and algebraic contexts.

Findings

Classification into three series: principal, unipotent, special

Any finitely generated involutive commutative two-valued group is isomorphic to one in these series

Results for topological and algebraic cases in specific dimensions

Abstract

A complete classification of finitely generated involutive commutative two-valued groups is obtained. Three series of such two-valued groups are constructed: principal, unipotent and special, and it is shown that any finitely generated involutive commutative two-valued group is isomorphic to a two-valued group belonging to one of these series. A number of classification results are obtained for topological involutive commutative two-valued groups in the Hausdorff and locally compact cases. The classification of algebraic involutive two-valued groups in the one-dimensional case is also discussed.