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Approximate roots of a valuation and the Pierce-Birkhoff Conjecture
This paper is a step in our program for proving the Piece-Birkhoff Conjecture
for regular rings of any dimension (this would contain, in particular, the
classical Pierce-Birkhoff conjecture which deals with polynomial rings over a
real closed field). We first recall the Connectedness and the Definable
Connectedness conjectures, both of which imply the Pierce - Birkhoff
conjecture. Then we introduce the notion of a system of approximate roots of a
valuation v on a ring A (that is, a collection Q of elements of A such that
every v-ideal is generated by products of elements of Q). We use approximate
roots to give explicit formulae for sets in the real spectrum of A which we
strongly believe to satisfy the conclusion of the Definable Connectedness
conjecture. We prove this claim in the special case of dimension 2. This proves
the Pierce-Birkhoff conjecture for arbitrary regular 2-dimensional rings
The connected Vietoris powerlocale
The connected Vietoris powerlocale is defined as a strong monad Vc on the category of locales. VcX is a sublocale of Johnstone's Vietoris powerlocale VX, a localic analogue of the Vietoris hyperspace, and its points correspond to the weakly semifitted sublocales of X that are âstrongly connectedâ. A product map Ă:VcXĂVcYâVc(XĂY) shows that the product of two strongly connected sublocales is strongly connected. If X is locally connected then VcX is overt. For the localic completion of a generalized metric space Y, the points of are certain Cauchy filters of formal balls for the finite power set with respect to a Vietoris metric. \ud
Application to the point-free real line gives a choice-free constructive version of the Intermediate Value Theorem and Rolle's Theorem. \ud
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The work is topos-valid (assuming natural numbers object). Vc is a geometric constructio
Connectedness of the space of smooth actions of on the interval
We prove that the spaces of \Cinf orientation-preserving actions of
on and nonfree actions of on the circle are connected
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