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Given a directed-complete partial order of finitely presented groups, I want to say that the free group is the bottom, but I don't think that is right.

Can anyone say what is a typical bottom in a directed-complete partial order of groups? Furthermore, are the finitely presentable groups the finite or compact elements?

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4  
You should probably clarify what you mean by a typical dcpo of groups. Otherwise the bottom can be anything you want: just take the dcpo {G}... – François G. Dorais Feb 24 '11 at 23:32
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Also, signatures (and greetings) aren't really necessary on MO. A signature is automatically attached to your posts. – François G. Dorais Feb 24 '11 at 23:34
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What does "dcpo" mean? – aaron Feb 25 '11 at 0:08
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Well, it's maximal if you think that a subgroup is greater than the group... which seems like an odd convention. – HJRW Feb 25 '11 at 6:01
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@SeanLawton your change to the title has actually made it incorrect. One should speak of a DCPO of groups, not of "DCPO groups". Moreover, why not insert the definite or indefinite article before the word "bottom" if you feel these changes are so necessary as to warrant bumping this question? – Yemon Choi Jun 4 at 14:02

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