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BOHR COMPACTIFICATIONS OF GROUPS AND RINGS
- Part of
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- Journal:
- The Journal of Symbolic Logic / Volume 88 / Issue 3 / September 2023
- Published online by Cambridge University Press:
- 04 February 2022, pp. 1103-1137
- Print publication:
- September 2023
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- Article
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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
${\mathrm {UT}}_3({\mathbb {Z}})$, the continuous Heisenberg group 
${\mathrm {UT}}_3({\mathbb {R}})$, and, more generally, groups of upper unitriangular and invertible upper triangular matrices over unital rings.
