Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
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 $A\to A$
unless I assume that $A\to A$ 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 $0 < 0$ 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 $\mathbb{Z}$ to $\mathbb{R}$. In order for $\mathbb{R}$ to be terminal, you’d need exactly one.
moved material on the Archimedean property in general to its own page Archimedean property
Anonymous
1 to 9 of 9