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A question for Recursion Theorists.

If you take a partial recursive function phi and two sets I and J such that
 dom(phi) contains I and range(phi) is included in J, you can define an
 obvious set-theoretic function f:I->J.

The question is: does EVERY surjective f admit a section, i.e. a g:J->I
 obtained in the same way from a partial recursive function, say psi, such
 that
	for each j in J    f(g(j))=j       ?

People-who-know will see I'm actually defining a category and asking "does
 every good (i.e surjective) epi split?"

I suspect this to be false, but I can't manage to find a counter-example; the
 difficulties arise from the clumsiness of restricting arbitrarily the domain
 of a partial recursive function.

Thanks in advance, and feel free to e-mail me!!

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	Roberto R. Chinnici	<chinnici@hp2.sm.dsi.unimi.it>

	Dipartimento di Scienze dell'Informazione
	Universita' degli studi di Milano

	"but now
	 she stands upon southampton dock
	 with her handkerchief
	 and her summer frock clings
	 to her wet body in the rain
	 in quiet desperation knuckles
	 white upon the slippery reins
	 she bravely waves the boys goodbye again"

				southampton dock, Pink Floyd
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