Two remarks on the first-order theories of Baumslag-Solitar groups

We characterize all finitely generated groups elementarily equivalent to a solvable Baumslag-Solitar group BS(m, 1). It turns out that a finitely generated group G is elementarily equivalent to BS(m, 1) if and only if G is isomorphic to BS(m, 1). Furthermore, we show that two Baumslag-Solitar groups are existentially (universally) equivalent if and only if they are elementarily equivalent if and only if they are isomorphic.     

Lipschitz monoids and Vahlen matrices

Recent studies of Vahlen matrices have again pointed out a very interesting multiplicative monoid that is present in every Clifford algebra; this motivates a new presentation of the knowledge already collected about this monoid before it got involved in Vahlen matrices. This monoid is first studied for itself (and under rather weak hypotheses), and secondly in view of a more effective application to Vahlen matrices. The “invariance property” of this monoid opens the way to new information about these matrices.     

Positive theories of free monoids

Translated fromAlgebra i Logika, Vol. 32, No. 2, pp. 148–159, March–April, 1993.     

Decidability of elementary theories of lattices of subsemigroups

Translated from Sibirskii Matematicheskii Zhurnal, Vol. 31, No. 1, pp. 27–33, January–February, 1990.     

Decidability boundaries for some classes of nilpotent and solvable groups

For any natural k ≥ 3 and l ≥ 2, we describe decidability boundaries for two varieties: the variety of all k-nilpotent groups and the variety of all l-solvable groups. Translated fromAlgebra i Logika, Vol. 39, No. 2, pp. 127–133, March–April, 2000.     

Equational theories as congruences of enriched monoids

The problem of characterizing the lattices of equational theories is still unsolved. In this paper we describe a class of monoids enriched by two unary operations and show that a lattice L is a lattice of equational theories if and only if L is isomorphic to a lattice of congruences of some enriched monoid belonging to . The author was supported by INTAS grant 03-51-4110 and The Alexander von Humboldt Foundation.     

Generalized Bowen-Franks groups of integral matrices with the same zeta function

In this paper the relation between the zeta function of an integral matrix and its generalized Bowen-Franks groups is studied. Suppose that A and B are nonnegative integral matrices whose invertible part is diagonalizable over the field of complex numbers and A and B have the same zeta function. Then there is an integer m, which depends only on the zeta function, such that, for any prime q such that gcd(q,m)=1, for any g(x)∈Z[x] with g(0)=1, the q-Sylow subgroup of the generalized Bowen-Franks group BF_g(x)(A) and BF_g(x)(B) are the same. In particular, if m=1, then zeta function determines generalized Bowen-Franks groups.     

Quasi-equational bases for graphs of semigroups,monoids and groups

The graph of an algebra A is the relational structure G(A) in which the relations are the graphs of the basic operations of A. Let denote by the class of all graphs of algebras from a class . We prove that if is a class of semigroups possessing a nontrivial member with a neutral element, then does not have finite quasi-equational basis. We deduce that, for a class of monoids or groups with a nontrivial member, also does not have finite quasi-equational basis.     

The classification of maximal inverse monoids of matrices

Semigroup Forum - Let $${\mathbb {F}}$$ be a field. The set $$M_{n}({\mathbb {F}})$$ of all $$n \times n$$ matrices over $${\mathbb {F}}$$ forms a monoid under usual matrix multiplication. We...     

Partial symmetry, reflection monoids and Coxeter groups

This is the first of a series of papers in which we initiate and develop the theory of reflection monoids, motivated by the theory of reflection groups. The main results identify a number of important inverse semigroups as reflection monoids, introduce new examples, and determine their orders.     

Sums of integral matrices

A study is begun of the behavior of invariant factors when integral matrices are added.     

Pairs,sets and sequences in first-order theories

In this paper we study the idea of theories with containers, like sets, pairs, sequences. We provide a modest framework to study such theories. We prove two concrete results. First, we show that first-order theories of finite signature that have functional non-surjective ordered pairing are definitionally equivalent to extensions in the same language of the basic theory of non-surjective ordered pairing. Second, we show that a first-order theory of finite signature is sequential (is a theory of sequences) iff it is definitionally equivalent to an extension in the same language of a system of weak set theory called WS.