Submission date: 9 November 2020

Abstract:

We introduce and study model-theoretic connected components of rings as an analogue of model-theoretic connected components of definable groups. We develop their basic theory and use them to describe both the definable and classical Bohr compactifications of rings. We then use model-theoretic connected components to explicitly calculate Bohr compactifications of some classical matrix groups, such as the discrete Heisenberg group UT_3(Z), the continuous Heisenberg group UT_3(R), and, more generally, groups of upper unitriangular and invertible upper triangular matrices over unital rings.

Mathematics Subject Classification: 03C98, 03C60, 20A15, 20G15, 16B70, 54H11, 03C45

Keywords and phrases: Bohr compactification, Heisenberg group, group of upper unitriangular matrices, model-theoretic connected components.

Full text arXiv 2011.04822: pdf, ps.