Stable structures homogeneous for a finite binary language

LetL be a finite relational language andH(L) denote the class of all countable stableL-structuresM for which Th(M) admits elimination of quantifiers. ForM ∈H(L) define the rank ofM to be the maximum value of CR(p, 2), wherep is a complete 1-type over Ø and CR(p, 2) is Shelah’s complete rank. IfL has only unary and binary relation symbols there is a uniform finite bound for the rank ofM ∈H(L). This theorem confirms part of a conjecture of the first author. Intuitively it says that for eachL there is a finite bound on the complexity of the structures inH(L).     

On countable homogeneous 3-hypergraphs

Summary We present some results on countable homogeneous 3-hypergraphs. In particular, we show that there is no unexpected homogeneous 3-hypergraph determined by a single constraint.The authors are grateful for support from NSERC (Canada) Grant A3040     

On modules for which the finite exchange property implies the countable exchange property

It has been a long standing open problem whether the finite exchange property implies the full exchange property for an arbitrary module. The main results of this paper are Theorem 1.1: For modules whose idempotent endomorphisms are central, the finite exchange property implies the countable exchange property, and Theorem 2.11: Over a ring with ace on essential right ideals, the finite exchange property implies the full exchange property for every quasi-continuous module. The latter can be viewed as a partial affirmative answer to an open problem of Mohamed and Muller [8].     

The conjugacy problem for automorphism groups of countable homogeneous structures

We consider the conjugacy problem for the automorphism groups of a number of countable homogeneous structures. In each case we find the precise complexity of the conjugacy relation in the sense of Borel reducibility.     

The number of finite relational structures

An elementary proof is presented of an asymptotic estimate for the number (up to isomorphism) of finite relational structures, under a quite general definition of “relational structure”.     

On finite lattices which are embeddable in subsemigroup lattices

Communicated by L. N. Shevrin     

Sets in a euclidean space which are not a countable union of convex subsets

We prove that if a closed planar setS is not a countable union of convex subsets, then exactly one of the following holds:

On some locally finite groups which are sharply triply transitive

We classify the groups satisfying the following conditions: i) is locally finite; ii) is a sharply triply transitive permutation group; iii) all elements of have fixpoints.Published in Ukraninskii Matematicheskii Zhurnal, Vol. 43, Nos. 7 and 8, pp. 1060–1065, July–August, 1991.     

On the relational complexity of a finite permutation group

The relational complexity \(\rho (X,G)\) of a finite permutation group is the least k for which the group can be viewed as an automorphism group acting naturally on a homogeneous relational system whose relations are k-ary (an explicit permutation group theoretic version of this definition is also given). In the context of primitive permutation groups, the natural questions are (a) rough estimates, or (preferably) precise values for \(\rho \) in natural cases; and (b) a rough determination of the primitive permutation groups with \(\rho \) either very small (bounded) or very large (much larger than the logarithm of the degree). The rough version of (a) is relevant to (b). Our main result is an explicit characterization of the binary (\(\rho =2\)) primitive affine permutation groups. We also compute the precise relational complexity of \({{\mathrm{Alt}}}_n\) acting on k-sets, correcting (Cherlin in Sporadic homogeneous structures. In: The Gelfand Mathematical Seminars, 1996–1999, pp. 15–48, Birkhäuser 2000, Example 5).     

The Lichtenbaum-Hartshorne theorem for modules which are finite over a ring homomorphism

Let φ:(R,m)→S be a flat ring homomorphism such that mS≠S. Assume that M is a finitely generated S-module with dim_R(M)=d. If the set of support of M has a special property, then it is shown that if and only if for each prime ideal satisfying , we have . This gives a generalization of the Lichtenbaum-Hartshorne vanishing theorem for modules which are finite over a ring homomorphism. Furthermore, we provide two extensions of Grothendieck’s non-vanishing theorem. Applications to connectedness properties of the support are given.     

On finite reflexive homomorphism-homogeneous binary relational systems

A structure is called homogeneous if every isomorphism between finitely induced substructures of the structure extends to an automorphism of the structure. Recently, P. J. Cameron and J. Nešet?il introduced a relaxed version of homogeneity: we say that a structure is homomorphism-homogeneous if every homomorphism between finitely induced substructures of the structure extends to an endomorphism of the structure.In this paper, we consider finite homomorphism-homogeneous relational systems with one reflexive binary relation. We show that for a large part of such relational systems (bidirectionally connected digraphs; a digraph is bidirectionally connected if each of its connected components can be traversed by ?-paths) the problem of deciding whether the system is homomorphism-homogeneous is coNP-complete. Consequently, for this class of relational systems there is no polynomially computable characterization (unless P=NP). On the other hand, in case of bidirectionally disconnected digraphs we present the full characterization. Our main result states that if a digraph is bidirectionally disconnected, then it is homomorphism-homogeneous if and only if it is either a finite homomorphism-homogeneous quasiorder, or an inflation of a homomorphism-homogeneous digraph with involution (a specific class of digraphs introduced later in the paper), or an inflation of a digraph whose only connected components are and .     

On almost complex structures which are not compatible with symplectic forms

In this Note we prove that the underlying almost complex structure to a non-Kähler almost Hermitian structure admitting a compatible connection with skew-symmetric torsion cannot be calibrated by a symplectic form even locally.     

Homological identities for a finite homogeneous graph

An application of the Atiyah-Bott trace identity to the study of the spectral characteristics of a finite (q+1)-homogeneous factorgraph Y=Γ/X is given (X is an infinite (q+1)-homogeneous tree, Γ a free group of isometries of X). Bibliography: 9 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 205, 1993, pp. 110–121.