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We construct a homeomorphism between the compact regular locale of integrals on a Riesz space and the locale of (valuations) on its spectrum. In fact, we construct two geometric theories and show that they are biinterpretable. The constructions are elementary and tightly connected to the Riesz space structure.
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We prove that every non-trivial valuation on an infinite superrosy field of positive characteristic has divisible value group and algebraically closed residue field. In fact, we prove the following more general result. Let $K$ be a field such that for every finite extension $L$ of $K$ and for every natural number $n>0$ the index $[L^*:(L^*)^n]$ is finite and, if $char(K)=p>0$ and $f: L to L$ is given by $f(x)=x^p-x$, the index $[L^+:f[L]]$ is also finite. Then either there is a non-trivial definable valuation on $K$, or every non-trivial valuation on $K$ has divisible value group and, if $char(K)>0$, it has algebraically closed residue field. In the zero characteristic case, we get some partial results of this kind. We also notice that minimal fields have the property that every non-trivial valuation has divisible value group and algebraically closed residue field.
This is a short introductory course to Set Theory, based on axioms of von Neumann--Bernays--Godel (briefly NBG). The text can be used as a base for a lecture course in Foundations of Mathematics, and contains a reasonable minimum which a good (post-graduate) student in Mathematics should know about foundations of this science.
We give two concrete examples of continuous valuations on dcpos to separate minimal valuations, point-continuous valuations and continuous valuations: (1) Let $mathcal J$ be the Johnstones non-sober dcpo, and $mu$ be the continuous valuation on $mathcal J$ with $mu(U) =1$ for nonempty Scott opens $U$ and $mu(U) = 0$ for $U=emptyset$. Then $mu$ is a point-continuous valuation on $mathcal J$ that is not minimal. (2) Lebesgue measure extends to a measure on the Sorgenfrey line $mathbb R_{l}$. Its restriction to the open subsets of $mathbb R_{l}$ is a continuous valuation $lambda$. Then its image valuation $overlinelambda$ through the embedding of $mathbb R_{l}$ into its Smyth powerdomain $mathcal Qmathbb R_{l}$ in the Scott topology is a continuous valuation that is not point-continuous. We believe that our construction $overlinelambda$ might be useful in giving counterexamples displaying the failure of the general Fubini-type equations on dcpos.
We denote by Conc(L) the semilattice of all finitely generated congruences of a lattice L. For varieties (i.e., equational classes) V and W of lattices such that V is contained neither in W nor its dual, and such that every simple member of W contains a prime interval, we prove that there exists a bounded lattice A in V with at most aleph 2 elements such that Conc(A) is not isomorphic to Conc(B) for any B in W. The bound aleph 2 is optimal. As a corollary of our results, there are continuum many congruence classes of locally finite varieties of (bounded) modular lattices.
We study the extension of the Kechris-Solecki-Todorcevic dichotomy on analytic graphs to dimensions higher than 2. We prove that the extension is possible in any dimension, finite or infinite. The original proof works in the case of the finite dimension. We first prove that the natural extension does not work in the case of the infinite dimension, for the notion of continuous homomorphism used in the original theorem. Then we solve the problem in the case of the infinite dimension. Finally, we prove that the natural extension works in the case of the infinite dimension, but for the notion of Baire-measurable homomorphism.
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