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I was looking at this old question

Hopf algebra and group structure correspondence for algebraic varieties

which says that there exists an equivalence between algebraic group structures on an algebraic variety $V$ (over a field) and Hopf algebraic structures on its algebra of regular functions $O(V)$. I would guess that this extends to an equivalence between algebraic semi-groups structures on $V$ and coalgebra structures on $O(V)$. Am I correct here?

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Yes. If your coalgebras have counits, then you are talking about algebraic monoids. –  Benjamin Steinberg Oct 4 '12 at 18:17
    
Yes, of course, these conditions are assumed. –  Dyke Acland Oct 4 '12 at 18:22
    
You might look at the books by Renner and Putcha on algebraic monoids. –  Benjamin Steinberg Oct 4 '12 at 18:23
    
P.S. Where does the proof of this come from? I would use the Hilbert Nullstellensatz, but doesn't that require K to be closed? –  Dyke Acland Oct 4 '12 at 18:43
    
Normally these guys work in the algebraically closed case and I assumed you were, too. If your variety is affine, there is no problem. The functor of points gives you a functor from commutative rings to monoids and hence the representing object will be a coalgebra. In the general case, without algebraically closed, I defer you to actual experts. –  Benjamin Steinberg Oct 4 '12 at 18:59
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