Primitive connected theories

We prove the quantifier-elimination theorem for so-called primitive connected theories, exemplified by theories of modules. The theorem generalizes the well-known Baur-Monk-Garavaglia theorem on the elimination of quantifiers in the model theory of modules. The definition of a class of primitive connected theories, as distinct from modules. is not supposed to impose any conditions on a type of axioms that would specify those theories. Dedicated to the 60th birthday of Academician Yu. L. Ershov Supported by RFFR grant No. 99-01-00600. Translated fromAlgebra i Logika, Vol. 39, No. 2, pp. 145–169, March–April, 2000.     

Finding model-companions via the skeleton

We present an analysis on the existentially closed (e.c.) structures for some theoryT in a rather complete categorical setting. The central notion of the skeleton ofT is defined. We formulate conditions on the skeleton which limit the number of e.c. structures forT, thereby ensuring the existence of a model-companion ofT. A new (purely categorical) proof of the uniqueness of the atomic structure is given for theories having the joint-embedding-property (JEP).As an application it is shown that a finitely generated universal Horn class possesses a model-companion — a resuilt that was proved earlier by a different method.Presented by Stanley Burris.     

Enveloping ringoids

We present notions of module over a universal algebra, and linear representation of a universal algebra, which have gained currency with categorical algebraists, we give several intriguing examples of these objects, showing that important aspects of universal algebras can be studied in this context, and we describe the theory ofenveloping ringoids, which have categories of left modules equivalent to corresponding categories of modules and linear representations. Algebras in many of the familiar varieties of algebras, which have underlying groups, turn out to have enveloping ringoids that are equivalent to familiar rings. Nonempty algebras in an abelian varietyV have an enveloping ringoid equivalent toR(V), the so-calledring of the variety V.Dedicated to the memory of Alan DayPresented by J. Sichler.     

A semi-abelian extension of a theorem by Takeuchi

We prove that the category of cocommutative Hopf algebras over a field is a semi-abelian category. This result extends a previous special case of it, based on the Milnor–Moore theorem, where the field was assumed to have zero characteristic. Takeuchi's theorem asserting that the category of commutative and cocommutative Hopf algebras over a field is abelian immediately follows from this new observation. We also prove that the category of cocommutative Hopf algebras over a field is action representable. We make some new observations concerning the categorical commutator of normal Hopf subalgebras, and this leads to the proof that two definitions of crossed modules of cocommutative Hopf algebras are equivalent in this context.     

Antiadditive Primitive Connected Theories

Our main goal is to prove that an infinite group is interpreted in every primitive connected non-superstable theory. Previously, we have introduced the concept of primitive connected theories, for which the quantifier elimination theorem was proved generalizing a similar elimination result for modules due to Baur, Monk, and Garavaglia. Here, we study primitive connected theories in which an infinite group is not interpreted, that is, theories that differ radically from theories of modules, but have a similar structure theory. Such are said to be antiadditive. (Note that theories of modules, as distinct from antiadditive ones, may be non-superstable.)     

Specht modules with abelian vertices

In this article, we consider indecomposable Specht modules with abelian vertices. We show that the corresponding partitions are necessarily p ²-cores where p is the characteristic of the underlying field. Furthermore, in the case of p≥3, or p=2 and μ is 2-regular, we show that the complexity of the Specht module S ^μ is precisely the p-weight of the partition μ. In the latter case, we classify Specht modules with abelian vertices. For some applications of the above results, we extend a result of M. Wildon and compute the vertices of the Specht module S^(pp)S^{(p^{p})} for p≥3.     

Azumaya Categories

We define the notions of Azumaya category and Brauer group in category theory enriched over some very general base category V. We prove the equivalence of various definitions, in particular in terms of separable categories or progenerating bimodules. When V is the category of modules over a commutative ring R with unit, we recapture the classical notions of Azumaya algebra and Brauer group and provide an alternative, purely categorical treatment of those theories. But our theory applies as well to the cases of topological, metric or Banach modules, to the sheaves of such structures or graded such structures, and many other examples.     

Partial Densities on Locally Compact Abelian Groups and Uniformly Distributed Sequences

 We investigate conditions under which a partial density on a locally compact abelian group can be extended to a density. The results allow applications to the theory of uniform distribution of sequences in locally compact abelian groups. Received August 27, 2001 Published online July 12, 2002     

Universal Horn classes and antivarieties of algebraic systems

We define and study universal Horn classes dual to varieties in both the syntactic and the semantic sense. Such classes, which we call antivarieties, appear naturally, e.g., in graph theory and in formal language theory. The basic results are the characterization theorem for antivarieties, the theorem on cores in axiomatizable color-families, and the decidability theorem for universal theories of families of interpretations of formal languages. Supported by RFFR grants Nos. 99-01-000485 and 96-01-00097, and also by DFG grant No. 436113/2670. Translated fromAlgebra i Logika, Vol. 39, No. 1, pp. 3–22, January–February, 2000.     

A wavelet theory for local fields and related groups

Let G be a locally compact abelian group with compact open subgroup H. The best known example of such a group is G = ℚ_p, the field of padic rational numbers (as a group under addition), which has compact open subgroup H = ℤ_p, the ring of padic integers. Classical wavelet theories, which require a non trivial discrete subgroup for translations, do not apply to G, which may not have such a subgroup. A wavelet theory is developed on G using coset representatives of the discrete quotient Ĝ/H^⊥ to circumvent this limitation. Wavelet bases are constructed by means of an iterative method giving rise to socalled wavelet sets in the dual group Ĝ. Although the Haar and Shannon wavelets are naturally antipodal in the Euclidean setting, it is observed that their analogues for G are equivalent.     

On the structure theory of the Iwasawa algebra of a p-adic Lie group

This paper is motivated by the question whether there is a nice structure theory of finitely generated modules over the Iwasawa algebra, i.e. the completed group algebra, Λ of a p-adic analytic group G. For G without any p-torsion element we prove that Λ is an Auslander regular ring. This result enables us to give a good definition of the notion of a pseudo-nullΛ-module. This is classical when G=ℤ ^k _p for some integer k≥1, but was previously unknown in the non-commutative case. Then the category of Λ-modules up to pseudo-isomorphisms is studied and we obtain a weak structure theorem for the ℤ _p -torsion part of a finitely generated Λ-module. We also prove a local duality theorem and a version of Auslander-Buchsbaum equality. The arithmetic applications to the Iwasawa theory of abelian varieties are published elsewhere. Received May 12, 2001 / final version received July 5, 2001?Published online September 3, 2001     

MV-algebras with operators (the commutative and the non-commutative case)

In the present paper we define the (pseudo) MV-algebras with n-ary operators, generalizing MV-modules and product MV-algebras. Our main results assert that there are bijective correspondences between the operators defined on a pseudo MV-algebra and the operators defined on the corresponding ℓ-group. We also provide a categorical framework and we prove the analogue of Mundici's categorical equivalence between MV-algebras and abelian ℓ-groups with strong unit. Thus, the category of pseudo MV-algebras with operators is equivalent to some category of ℓ-groups with operators.     

Congruences between derivatives of geometric <Emphasis Type="Italic">L</Emphasis>-functions

We prove a natural refinement of a theorem of Lichtenbaum describing the leading terms of Zeta functions of curves over finite fields in terms of Weil-étale cohomology. We then use this result to prove the validity of Chinburg’s Ω(3)-Conjecture for all abelian extensions of global function fields, to prove natural refinements and generalisations of the refined Stark conjectures formulated by, amongst others, Gross, Tate, Rubin and Popescu, to prove a variety of explicit restrictions on the Galois module structure of unit groups and divisor class groups and to describe explicitly the Fitting ideals of certain Weil-étale cohomology groups. In an Appendix coauthored with K.F. Lai and K.-S. Tan we also show that the main conjectures of geometric Iwasawa theory can be proved without using either crystalline cohomology or Drinfeld modules.     

The homotopy categorical crossed module of a CW-complex

Crossed modules have longstanding uses in homotopy theory and the cohomology of groups. The corresponding notion in the setting of categorical groups, that is, categorical crosses modules, allowed the development of a low-dimensional categorical group cohomology. Now, its relevance is also shown here to homotopy types by associating, to any pointed CW-complex (X,∗), a categorical crossed module that algebraically represents the homotopy 3-type of X.     

Jaffard-Ohm correspondence and Hochster duality

We study categorical aspects of the Jaffard–Ohm correspondencebetween abelian l-groups and Bézout domains and showthat this correspondence is close to a localization. For thispurpose, we establish a general extension theorem for valuationswith value group that is an abelian l-group. As an application,we prove Anderson's conjecture which refines the Jaffard–Ohmcorrespondence. We then extend the correspondence to sheaveson spectral spaces and show that the spectrum of a Bézoutdomain and the spectrum of its corresponding abelian l-groupprovide a concrete example for Hochster's duality of spectralspaces.     

Weak Equivalence of Internal Categories

Weak equivalence is defined as equivalence in the bicategory of modules between internal categories. It is known that two categories are weakly equivalent if and only if their Cauchy completions are equivalent. We prove that this condition can be generalized to a suitable notion of intermediate category, stable under composition with weak equivalences. Applications to categorical Morita theory are given.     

关于余极限范畴(英文)

本文研究了余极限范畴.利用余完备Abel范畴的定义,证明了余完备Abel范畴A的余极限范畴Acl是余完备的Abel范畴,并得到一类等价于模范畴的余极限范畴,从而推广了文献[9]中的一些结果.     

On decidability of the decomposability problem for finite theories

We consider the decomposability problem for elementary theories, i.e. the problem of deciding whether a theory has a nontrivial representation as a union of two (or several) theories in disjoint signatures. For finite universal Horn theories, we prove that the decomposability problem is $ \sum _1^0 $ \sum _1^0 -complete and, thus, undecidable. We also demonstrate that the decomposability problem is decidable for finite theories in signatures consisting only of monadic predicates and constants.