Affine algebraic sets relative to an algebraic theory

For an arbitrary algebraic theory T and a prescribed T-algebraL, the geometrical category AfAlgSet(L) of affine algebraic sets over the affine lineL is build up. It is proved to be dually equivalent to the category FcAlg(L) of functional T-algebras overL. The Galois theory ofL is defined and a Galois duality established. These geometrical categories are characterized up to an equivalence of categories.     

Generalized analytic sets

We study completely valued algebraically closed fields in languages with integral analytic functions. it is shown that a nontrivial model theory of integral functions can be developed on a condition that a language is chosen properly. Supported by ISF grant No. RKE300. Translated from Algebra i Logika, Vol. 36, No. 4, pp. 387–406, July–August, 1997.     

All compact manifolds are homeomorphic to totally algebraic real algebraic sets

Both authors supported in part by the N.S.F.     

Categories of algebraic sets

The category of algebraic sets is defined in a straightforward way for any algebraic theory . It is a concrete, complete and cocomplete category dually equivalent to a full reflective subcategory of the category of -algebras. For the algebraic theory of commutative algebras over a field K, we get the algebraic sets over K.     

Weakly analytic sets for function spaces

We define and study weakly analytic sets for function spaces. This concept generalizes the concept of weak analyticity for function algebras. We also define weak analyticity for the space of affine functions using the ideas of convexity. We show that it coincides with the same concept regarding it as a function space. We give several examples to illustrate the concepts.     

Generating sets for rings of algebraic integers

Let R be the ring of integers of some finite algebraic extension of the rationals Q of degree n. A necessary and sufficient condition for s elements of R to be an R-basis is given, in terms of the Hermite normal form of a certain n ×ns integral matrix depending on the elements, and on the structure constants of R.     

Criteria for the injectivity of analytic sheaves

It is shown that a moduleL over the sheafO of germs of holomorphic functions on a domain G of Cⁿ is injective if and only if the following conditions are satisfied; a)L is flabby; b) for every closed set S ?G and every point z λ G, the stalk se _z of the sheafS ^L;U1→Γ S ^(U:L) is an injectiveO _z -module. It follows in particular that the sheaf of germs of hyperfunctions is injective over the sheaf of germs of analytic functions.     

Bernstein sets with algebraic properties

We construct Bernstein sets in ℝ having some additional algebraic properties. In particular, solving a problem of Kraszewski, Rałowski, Szczepaniak and Żeberski, we construct a Bernstein set which is a < c-covering and improve some other results of Rałowski, Szczepaniak and Żeberski on nonmeasurable sets.     

Homotopy properties of algebraic sets

In this paper there are announced several homotopy theorems on algebraic subsets of complex projective spaces. Some of the theorems generalize and refine results of the preceding note of the author (RZhMat, 1978, 5A531). It is also asserted that any -dimensional complex affine algebraic set has the homotopy type of a finite -dimensional cellular space. A generalization is given, in two versions, of a theorem of Bart-Larsen (RZhMat, 1973, 7A587).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 83, pp. 67–72, 1979.     

A sufficient condition for a group automorphism to be algebraic

Translated from Algebra i Logika, Vol. 28, No. 1, pp. 117–119, January–February, 1989.     

Vaught's conjecture on analytic sets

Let be a Polish group. We characterize when there is a Polish space with a continuous -action and an analytic set (that is, the Borel image of some Borel set in some Polish space) having uncountably many orbits but no perfect set of orbit inequivalent points.

Such a Polish -space and analytic exist exactly when there is a continuous, surjective homomorphism from a closed subgroup of onto the infinite symmetric group, , consisting of all permutations of equipped with the topology of pointwise convergence.

     

On approximation of analytic sets

We give conditions sufficient for the existence of a sequence of Nash sets convergent to a given analytic set. Moreover, for limit sets which are k-sheeted analytic covers we additionally require that sets in approximating sequence are also k-sheeted analytic covers. Then we present examples of applications of the developed theory.Research partially supported by KBN Grant no. 2P03A 015 22Mathematics Subject Classification (2000): 32E30, 32C07, 32C25