On the free automata and tensor product

In this paper we shall introduce the algebraic structure of a tensor product for arbitrarily given automata, giving a defintion of the tensor product for automata. We introduce and study that for any setX there always exists a free automaton onX. The existence of a tensor product for automata will be investigated in the same way like modules do.     

Non-free torsion-free profinite groups with open free subgroups

For every integere≧3 there exists a non-free torsion-free profinite group containing a as an open subgroup. Supported by Rothschild Fellowship.     

On stable torsion-free nilpotent groups

Summary We show that an infinite field is interpretable in a stable torsion-free nilpotent groupG of classk, k>1. Furthermore we prove thatG/Z _k-1 (G) must be divisible. By generalising methods of Belegradek we classify some stable torsion-free nilpotent groups modulo isomorphism and elementary equivalence.Supported by the Deutsche Forschungsgemeinschaft     

On multipliers of torsion-free nilpotent groups

The notion of the Schur multiplier is carried over to torsion-free nilpotent groups of finite rank, and the relation between the rank of a torsion-free nilpotent group and the rank of its multiplier is determined, [3].Translated from Matematicheskie Zametki, Vol. 5, No. 5, pp. 541–544, May, 1969.     

Probabilistic tree automata and context free languages

A notion of a probabilistic tree automaton is defined and a condition is given under which it is equivalent to a usual tree automaton. A theorem about context free languages is stated. This research was partially supported by ONR, Contract No. N00014 69 C 0192.     

On torsion-free groups with finite regular file bases

The following question was asked by V. V. Bludov in The Kourovka Notebook in 1995: If a torsion-free group has a finite system of generators , ..., such that every element of has a unique presentation in the form where , is it true that is virtually polycyclic? The answer is ``not always.' A counterexample is constructed in this paper as a group presented by generators and defining relations.

     

Some remarks on the structure of free automata

In this paper we define automata-linearly independence. An automatonM has a basis B iffM is free provided that we assume that the action ofS onX × S is (x,s)a = (x,sa) for alla, s ∈ S andx ∈X. If a semigroupS is PRID, every subautomaton of a freeS-automaton is free.     

On the ideals of torsion-free rings of rank one and two

Let A be a torsion-free Abelian group of rank one or two. We use the type set of A to give necessary and sufficient conditions for the subgroups of A to be ideals in every ring on A.     

On genus and embeddings of torsion-free nilpotent groups of class two

Summary We study embeddings between torsion-free nilpotent groups having isomorphic localizations. Firstly, we show that for finitely generated torsion-free nilpotent groups of nilpotency class 2, the property of having isomorphicP-localizations (whereP denotes any set of primes) is equivalent to the existence of mutual embeddings of finite index not divisible by any prime inP. We then focus on a certain family Γ of nilpotent groups whose Mislin genera can be identified with quotient sets of ideal class groups in quadratic fields. We show that the multiplication of equivalence classes of groups in Γ induced by the ideal class group structure can be described by means of certain pull-back diagrams reflecting the existence of enough embeddings between members of each Mislin genus. In this sense, the family Γ resembles the family N₀ of infinite, finitely generated nilpotent groups with finite commutator subgroup. We also show that, in further analogy with N₀, two groups in Γ with isomorphic localizations at every prime have isomorphic localizations at every finite set of primes. We supply counterexamples showing that this is not true in general, neither for finitely generated torsion-free nilpotent groups of class 2 nor for torsion-free abelian groups of finite rank. Supported by DGICYT grant PB94-0725 This article was processed by the author using the LAT_EX style filecljour1 from Springer-Verlag.     

On subdirect sums of Abelian torsion-free groups of rank 1

In this paper, we study torsion-free Abelian groups of rank 2, which are subdirect sums of two divisible rational groups, with the inducing group ℚ/ℤ. The class of special groups is defined and investigated. It is shown that there is a one-to-one correspondence between the set of all special groups and the multiplicative group of unity elements of the ring of universal numbers. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 3, pp. 209–222, 2007.     

On quasivarieties of axiomatic rank 3 of torsion-free nilpotent groups

We study the lattice of quasivarieties of axiomatic rank at most 3 of torsion-free nilpotent groups of class at most 3. We prove that this lattice has cardinality of the continuum and includes a sublattice that is order isomorphic to the set of real numbers. Also we establish that the lattice of quasivarieties of axiomatic rank at most 2 of these groups is a 5-element chain.