2-local Baer correspondence

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Definition

The 2-local Baer correspondence is a slight variation on the usual Baer correspondence that aims to establish a correspondence between things that 2-locally look like Baer Lie rings and things that 2-locally look like Baer Lie groups. Explicitly, it is a correspondence:

Baer alternating loop rings \leftrightarrow Baer diassociative loops

The correspondence:

Baer Lie rings \leftrightarrow Baer Lie groups

is a subcorrespondence of this correspondence.

Observations

The first thing we observe is that under this correspondence, it is possible for Lie rings that are not of class two (but have 2-local class two) to get mapped to diassociative loops that aren't groups. Conversely, it is possible for groups that are not of class two (but have 2-local class two) to get mapped to alternating rings that are not Lie rings.