Constructive Matrix and Orderable Groups

Abstract

We study into the relationship between constructivizations of an associative commutative ring K with unity and constructivizations of matrix groups GLn(K) (general), SLn(K) (special), and UTn(K) (unitriangular) over K. It is proved that for n≥ 3, a corresponding group is constructible iff so is K. We also look at constructivizations of ordered groups. It turns out that a torsion-free constructible Abelian group is orderly constructible. It is stated that the unitriangular matrix group UTn(K) over an orderly constructible commutative associative ring K is itself orderly constructible. A similar statement holds also for finitely generated nilpotent groups, and countable free nilpotent groups.