Vanishing evaluations of simple functors

The classification of simple biset functors is known, but the evaluation of a simple biset functor at a finite group G$G$ may be zero. We investigate various situations where this happens, as well as cases where this does not occur. We also prove a closed formula for such an evaluation under some restrictive conditions on  G$G$.     

Extension groups between simple Mackey functors

In order to better understand the structure of indecomposable projective Mackey functors, we study extension groups of degree 1 between simple Mackey functors. We explicitly determine these groups between simple functors indexed by distinct normal subgroups. We next study the conditions under which it is possible to restrict ourselves to that case, and we give methods for calculating extension groups between simple Mackey functors which are not indexed by normal subgroups. We then focus on the case where the simple Mackey functors are indexed by the same subgroup. In this case, the corresponding extension group can be embedded in an extension group between modules over a group algebra, and we describe the image of this embedding. In particular, we determine all extension groups between simple Mackey functors for a p-group and for a group that has a normal p-Sylow subgroup. Finally, we compute higher extension groups between simple Mackey functors for a group that has a p-Sylow subgroup of order p.     

Amalgamation properties and finite models in Ln-theories

 Djordjević [Dj 1] proved that under natural technical assumptions, if a complete L ⁿ -theory is stable and has amalgamation over sets, then it has arbitrarily large finite models. We extend his study and prove the existence of arbitrarily large finite models for classes of models of L ⁿ -theories (maybe omitting types) under weaker amalgamation properties. In particular our analysis covers the case of vector spaces. Received: 24 July 2000 / Published online: 20 December 2001     

Closure properties for fitting functors

Fitting functors have been introduced as a generalization of injectors and radicals for Fitting classes of finite (soluble) groups. This paper provides definitions and properties of certain types of Fitting functors which resemble injectors and radicals for Fitting classes with additional closure properties, most prominently subgroup- or quotient-closed Fitting classes and Fischer classes.     

On some properties of cohomotopy-type functors

It is shown that if II ⁿ ,n\s>2, are Chogoshvili's cohomotopy functors [1, 2, 3], then

On omega-categorical simple theories

In the present paper we shall prove that countable ω-categorical simple CM-trivial theories and countable ω-categorical simple theories with strong stable forking are low. In addition, we observe that simple theories of bounded finite weight are low.     

Strict polynomial functors and coherent functors

We build an explicit link between coherent functors in the sense of Auslander [Coherent functors. Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965). Springer, Berlin, pp. 189–231, 1966] and strict polynomial functors in the sense of Friedlander and Suslin (Invent. Math. 127(2), 209–270, 1997). Applications to functor cohomology are discussed. V. Franjou is partially supported by the LMJL—Laboratoire de Mathématiques Jean Leray, CNRS: Université de Nantes, école Centrale de Nantes, and acknowledges the hospitality and support of CRM Barcelona where the final corrections on this paper were implemented.     

On dividing chains in simple theories

Dividing chains have been used as conditions to isolate adequate subclasses of simple theories. In the first part of this paper we present an introduction to the area. We give an overview on fundamental notions and present proofs of some of the basic and well-known facts related to dividing chains in simple theories. In the second part we discuss various characterizations of the subclass of low theories. Our main theorem generalizes and slightly extends a well-known fact about the connection between dividing chains and Morley sequences (in our case: independent sequences). Moreover, we are able to give a proof that is shorter than the original one. This result motivates us to introduce a special property of formulas concerning independent dividing chains: For any dividing chain there exists an independent dividing chain of the same length. We study this property in the context of low, short and ω -categorical simple theories, outline some examples and define subclasses of low and short theories, which imply this property. The results give rise to further studies of the relationships between some subclasses of simple theories. Research supported by CNPq grant 150309/2003-1. Research supported by CNPq grant 304365/2003-3 (Modelos, Provas e Algoritmos)     

A primer of simple theories

 We present a self-contained exposition of the basic aspects of simple theories while developing the fundamentals of forking calculus. We expound also the deeper aspects of S. Shelah's 1980 paper Simple unstable theories. The concept of weak dividing has been replaced with that of forking. The exposition is from a contemporary perspective and takes into account contributions due to S. Buechler, E. Hrushovski, B. Kim, O. Lessmann, S. Shelah and A. Pillay. Received: 12 November 1999 / Revised version: 1 June 2001 / Published online: 12 July 2002     

Very simple theories without forking

We prove Vaught's conjecture for minimal trivial simple theories satisfying the generalized independence theorem. Research supported by KBN grant 2 P03A 006 09     

Certain homotopic properties of functors with finite supports

It is proved that Shchepin-normal functors with finite supports in the category of compacta are continuous in the topology of homotopically n-regular Kuratowski convergence in the space of LCⁿ-subsets of a metric compactum. These functors also preserve the property of mappings of polyhedra to induce an isomorphism of homotopic groups of dimension not greater than n.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 11, pp. 1618–1620, November, 1992.     

On geometric properties of functors of positive-homogenous and semiadditive functionals

We investigate the functor OH of positive-homogenous functionals and the functor OS of semiadditive functionals. We prove that OH(X) is an absolute retract if and only if X is an open-generated compactum, and OS(X) is an absolute retract if and only if X is an opengenerated compactum of weight ≤ ω₁. We investigate the softness of mappings of multiplication of monads generated by these functors.     

Adding Skolem functions to simple theories

We examine the conditions under which we can keep simplicity or categoricity after adding a Skolem function to the theory.Mathematics Subject Classification (2000): 03C45, 03C50     

Elementary theories of completely simple semigroups

The connections between first-order formulas over a completely simple semigroupC and corresponding formulas over its structure groupH are found in this paper. For the case of finite sandwich-matrix the criterion of decidability of the elementary theoryT(C) is established in terms of the elementary theory ofH in the enriched signature (Theorem 1). For the general case the criterion is established in terms of two-sorted algebraic systems (Theorem 2). Sufficient conditions in terms ofH for decidability and for undecidability ofT(C) are outlined. Corollaries and examples are presented, among them an example of a completely simple semigroup with a finite structure group and with undecidable elementary theory (Theorem 3).