Reflection to 0-dimensional locales

["George Janelidze" <George.Janelidze@uct.ac.za>]

Dear Colleagues,

Let me repeat from my exchange of massages with Steve Vickers:

> As you know, a locale is called 0-dimensional if all its elements are
> joins
> of complemented ones. By a morphism L--->L' of locales I shall mean a map
> L'--->L that preserves all joins and finite meets (as usually). The
> inclusion functor
>
> 0-Dimensional locales--->Locales
>
> has a left adjoint F, for which
>
> F(L)={x in L | x is a join of complementary elements}.
>
> Question: Is F semi-left-exact?
>
> I mentioned this question several times in past to several people... I am
> very interested to know the answer, no matter whether it is YES or NO; if
> NO, then I have weaker questions...

Almost immediately after writing this I received the following message:

"...a sufficient condition for F failing to be semi-left exact is for the
coproduct of a connected frame Q and a Boolean frame X to have a
complemented element that is not in the image of the inclusion of X. I
believe such an example is described in chapter XIII, section 4, pages
260--266 of the book Frames and Locales by Picado and Pultr..."

The author is Graham Manuell, a PhD student at the University of Edinburgh
who did his MSc in Cate Town.

I looked at the book: it will take me a long time (which I don't have now)
to check the details, because understanding them will require carefully
reading every word of those pages... But if what the book says is correct (I
cannot imagine these good authors to be careless of course!), then what
Graham says is certainly correct, in spite of the fact that
semi-left-exactness is not mentioned in the book. The example, as the
authors say, was found by I. Kriz (I apologize for not using proper accents
on r, i, and z).

Moreover, most of the "weaker questions" I had in mind, are also answered...

But I still have a question: Kriz's example is presented as a
counter-example to a frame-theoretic counterpart of a purely topological
property, but now - thanks to Graham's simple remark - it is also a
counter-example to semi-left-exactness, whose topological counterpart also
fail (unless we restrict spaces to, say, locally connected, or compact). Is
there an easier counter-example?

George Janelidze


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