TL;DR
This paper characterizes bounded automorphisms and endomorphisms in algebraic structures, showing they are definable or composed of specific endomorphisms in certain expanded fields.
Contribution
It provides a definability result for bounded automorphisms and describes the structure of bounded endomorphisms in expanded fields with Pfaffian families.
Findings
Bounded automorphisms are definable in certain algebraic structures.
Bounded endomorphisms in expanded fields are compositions of endomorphisms associated with the expansion.
In some cases, bounded endomorphisms involve powers of the Frobenius.
Abstract
A bounded automorphism of a field or a group with trivial approximate centre is definable. In an expansion of a field by a Pfaffian family F of additive endomorphisms such that algebraic closure in the expansion coincides with relative field-algebraic closure of the F-substructure generated, a bounded endomorphism, possibly composed with a power of the Frobenius, is a composition of endomorphisms associated with F.
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Taxonomy
TopicsRings, Modules, and Algebras · Advanced Topology and Set Theory · Homotopy and Cohomology in Algebraic Topology