313 research outputs found
Certain minimal varieties are set-theoretic complete intersections
We present a class of homogeneous ideals which are generated by monomials and
binomials of degree two and are set-theoretic complete intersections. This
class includes certain reducible varieties of minimal degree and, in
particular, the presentation ideals of the fiber cone algebras of monomial
varieties of codimension two
A universal algorithm for Krull's theorem
We give a computational interpretation to an abstract formulation of Krull's theorem, by analysing its classical proof based on Zorn's lemma. Our approach is inspired by proof theory, and uses a form of update recursion to replace the existence of maximal ideals. Our main result allows us to derive, in a uniform way, algorithms which compute witnesses for existential theorems in countable abstract algebra. We give a number of concrete examples of this phenomenon, including the prime ideal theorem and Krull's theorem on valuation rings
Translation invariant filters and van der Waerden's Theorem
We present a self-contained proof of a strong version of van der Waerden's
Theorem. By using translation invariant filters that are maximal with respect
to inclusion, a simple inductive argument shows the existence of "piecewise
syndetically"-many monochromatic arithmetic progressions of any length k in
every finite coloring of the natural numbers. All the presented constructions
are constructive in nature, in the sense that the involved maximal filters are
defined by recurrence on suitable countable algebras of sets. No use of the
axiom of choice or of Zorn's Lemma is needed
Translation Invariant Filters and van der Waerden’s Theorem
We present a self-contained proof of a strong version of van der Waerden’s Theorem. By using translation invariant filters that are maximal with respect to inclusion, a simple inductive argument shows the existence of “piecewise syndetically”-many monochromatic arithmetic progressions of any length k in every finite coloring of the natural numbers. All the presented constructions are constructive in nature, in the sense that the involved maximal filters are defined by recurrence on suitable countable algebras of sets. No use of the axiom of choice or of Zorn’s Lemma is needed
Apartness, sharp elements, and the Scott topology of domains
Working constructively, we study continuous directed complete posets (dcpos) and the Scott topology. Our two primary novelties are a notion of intrinsic apartness and a notion of sharp elements. Being apart is a positive formulation of being unequal, similar to how inhabitedness is a positive formulation of nonemptiness. To exemplify sharpness, we note that a lower real is sharp if and only if it is located. Our first main result is that for a large class of continuous dcpos, the Bridges-Vîţǎ apartness topology and the Scott topology coincide. Although we cannot expect a tight or cotransitive apartness on nontrivial dcpos, we prove that the intrinsic apartness is both tight and cotransitive when restricted to the sharp elements of a continuous dcpo. These include the strongly maximal elements, as studied by Smyth and Heckmann. We develop the theory of strongly maximal elements highlighting its connection to sharpness and the Lawson topology. Finally, we illustrate the intrinsic apartness, sharpness, and strong maximality by considering several natural examples of continuous dcpos: the Cantor and Baire domains, the partial Dedekind reals, the lower reals and, finally, an embedding of Cantor space into an exponential of lifted sets
04351 Abstracts Collection -- Spatial Representation: Discrete vs. Continuous Computational Models
From 22.08.04 to 27.08.04, the Dagstuhl Seminar 04351
``Spatial Representation: Discrete vs. Continuous Computational Models\u27\u27
was held in the International Conference and Research Center (IBFI),
Schloss Dagstuhl.
During the seminar, several participants presented their current
research, and ongoing work and open problems were discussed. Abstracts of
the presentations given during the seminar as well as abstracts of
seminar results and ideas are put together in this paper. The first section
describes the seminar topics and goals in general.
Links to extended abstracts or full papers are provided, if available
The Theory of Involutive Divisions and an Application to Hilbert Function Computations
AbstractGeneralising the divisibility relation of terms we introduce the lattice of so-called involutive divisions and define the admissibility of such an involutive division for a given set of terms. Based on this theory we present a new approach for building a general theory of involutive bases of polynomial ideals. In particular, we give algorithms for checking the involutive basis property and for completing an arbitrary basis to an involutive one. It turns out that our theory is more constructive and more flexible than the axiomatic approach to general involutive bases due to Gerdt and Blinkov.Finally, we show that an involutive basis contains more structural information about the ideal of leading terms than a Gröbner basis and that it is straightforward to compute the (affine) Hilbert function of an idealIfrom an arbitrary involutive basis of alI
Note on radical and prime E-ideals
We show that the ring of exponential polynomials is not Noetherian even
respect to prime E-ideals. Moreover we give a characterization of exponential
radical ideal
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