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The claim appears to be repeated at real number:
Alternatively, an archimedean group is terminal if it is the terminal object in the category of archimedean groups.
Hmm, this doesn’t make sense
The positive integers are embedded into the function group A→A
unless I assume that A→A is automorphisms of A. That is, unless A is abelian, somehow forced by the linearity of the ordering on A.
The Wikipedia article on archimedean groups does say that they’re abelian, but perhaps you’re right perhaps in that this ought to be said first. Perhaps some clean-up is in order.
Todd wrote:
Replaced the wrong sentence that the real numbers form a terminal archimedean group
Is that because, unlike in the case for fields, the trivial group can be trivially made into a linearly ordered group by declaring 0Unknown character0 to be false? So the terminal archimedean group would just be the trivial group?
Yes, of course.
Consider also that there are many maps of archimedean groups from ℤ to ℝ. In order for ℝ to be terminal, you’d need exactly one.
moved material on the Archimedean property in general to its own page Archimedean property
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