Higher Central Extensions in Mal’tsev Categories

Higher dimensional central extensions of groups were introduced by G. Janelidze as particular instances of the abstract notion of covering morphism from categorical Galois theory. More recently, the notion has been extended to and studied in arbitrary semi-abelian categories. Here, we further extend the scope to exact Mal’tsev categories and beyond. For this, we consider conditions on a Galois structure Γ = (?, ??, I, H, η, ?) which insure the existence of an induced Galois structure Γ ₁ = (?₁, ??₁, I ₁, H ₁, η ¹, ? ¹) such that ?₁ and ??₁ are full subcategories of the arrow category Arr(?) consisting, respectively, of all morphisms in the class ?, and of all covering morphisms with respect to Γ. Moreover, we prove that Γ ₁ satisfies the same conditions as Γ, so that, inductively, we obtain, for each n ≥ 1, a Galois structure Γ _n = (Γ _n?1)₁, whose coverings we call n + 1-fold central extensions.     

Centers of Mal’tsev and alternative algebras

Let ϕ be an associative commutative ring with unity containing 1/6. Let A and B be a free Mal’tsev and a free alternative ϕ-algebras on a set of k≥6 free generators, respectively. We construct nonzero homogeneous elements of degree 7 belonging to an annihilatorAnnA of A, and nonzero homogeneous elements of degree 7 belonging to the center Z(B) of B. It is shown that a nilpotent Mal’tsev algebra of index 8 on a set of 6 generators has no faithful representation. Supported by RFFR grant No. 96-01-01511, and by the Program “Universities of Russia: Fundamental Research.” Translated fromAlgebra i Logika, Vol. 38, No. 5, pp. 613–635, September–October, 1999.     

On distributive lattices and weakly Mal’tsev categories

We prove that a variety of lattices is weakly Mal’tsev if and only if it is a variety of distributive lattices.     

Congruence Distributivity in Goursat and Mal’cev Categories

We characterize the congruence distributive property for the Goursat and regular Malcev categories in terms of preservation of the intersection by direct images. This splits a previous characterization for the exact Malcev categories in two cases: plain congruence distributive property and weak congruence distributive property.Mathematics Subject Classifications (2000) Primary: 08B10, 08C05, 18C05; secondary: 08A30, 18D05.     

Admissible Galois Structures and Coverings in Regular Mal’cev Categories

Given a regular Gumm category such that any regular epimorphism is effective for descent, we prove that any Birkhoff subcategory in gives rise to an admissible Galois structure. This result allows one to consider some new applications of the categorical Galois theory in the context of topological algebras. Given a regular Mal’cev category , we first characterize the coverings of the Galois structure induced by the subcategory of the abelian objects in . Then we consider as a subcategory of the category of the equivalence relations in , and we characterize the coverings of the corresponding Galois structure . By composing the Galois structures and we obtain the Galois structure induced by as a subcategory of . We give the characterization of the -coverings in terms of the coverings of and .     

Strong Mal’tsev conditions in varieties of Abelian groups

Let Ab_d be a variety of Abelian groups of a finite exponent d≥1 and SC (Ab_d) be the set of all strong Mal’tsev conditions satisfied in Ab_d. We define the concept of a η-basis in SC(Ab_d) in terms of a basis w.r.t. a class η of varieties with commutative operations. The algorithm for constructing η-bases of any finite length in SC(Ab_d) is presented. For the variety Ab of all Abelian groups, we specify absolute bases of length 2 in SC(Ab) which are simultaneously η-bases. Bases of length 2 with similar properties are constructed also in SC(Ab_d), for any natural number d≥2. Translated fromAlgebra i Logika, Vol. 38, No. 6, pp. 723–742, November–December, 1999.     

All congruence lattice identities implying modularity have Mal’tsev conditions

For an arbitrary lattice identity implying modularity (or at least congruence modularity) a Maltsev condition is given such that the identity holds in congruence lattices of algebras of a variety if and only if the variety satisfies the corresponding Maltsev condition.     

A Mal’tsev basis for a partially commutative nilpotent metabelian group

We find a canonical representation for elements of a partially commutative group in a variety of soluble groups of derived length two and nilpotency class at most c ≥ 1.     

Closedness properties of internal relations V: Linear Mal’tsev conditions

As J. W. Snow showed, every linear Mal’tsev condition on a variety of universal algebras, is equivalent to a relational condition on . Using slightly different relational reformulations of linear Mal’tsev conditions, we develop a purely categorical approach to these conditions. Received August 10, 2006; accepted in final form January 23, 2007.     

The Best Constant in Siegel’s Lemma

We give a new proof of the basis form of Siegels Lemma over an algebraic number field k in which the field and dimension dependent constant is best possible. This constant is equal to a generalization of Hermites constant for the algebraic number field k that has recently been studied by J. L. Thunder.Research supported in part by the National Science Foundation (DMS-00-88915).Communicated by W. SchmidtReceived April 4, 2002; in revised form April 28, 2003 Published online August 28, 2003     

Siegel’s Lemma and Sum-Distinct Sets

Let L(x)=a ₁ x ₁+a ₂ x ₂+???+a _n x _n , n≥2, be a linear form with integer coefficients a ₁,a ₂,…,a _n which are not all zero. A basic problem is to determine nonzero integer vectors x such that L(x)=0, and the maximum norm ‖x‖ is relatively small compared with the size of the coefficients a ₁,a ₂,…,a _n . The main result of this paper asserts that there exist linearly independent vectors x ₁,…,x _n?1∈? ⁿ such that L(x _i )=0, i=1,…,n?1, and $$\|{\mathbf{x}}_{1}\|\cdots\|{\mathbf{x}}_{n-1}\|<\frac{\|{\mathbf{a}}\|}{\sigma_{n}},$$ where a=(a ₁,a ₂,…,a _n ) and $$\sigma_{n}=\frac{2}{\pi}\int_{0}^{\infty}\left(\frac{\sin t}{t}\right)^{n}\,dt.$$ This result also implies a new lower bound on the greatest element of a sum-distinct set of positive integers (Erdös–Moser problem). The main tools are the Minkowski theorem on successive minima and the Busemann theorem from convex geometry.     

Once more on periodic products of groups and on a problem of A. I. Mal’tsev

A new operation of product of groups, the n-periodic product of groups for odd exponent n ≥ 665, was proposed by the author in 1976 in the paper [1]. This operation is described on the basis of the Novikov-Adyan theory introduced in the monograph [2] of the author. It differs from the classic operations of direct and free products of groups, but has all of the natural properties of these operations, including the so-called hereditary property for subgroups. Thus, the well-known problem of A. I. Mal’tsev on the existence of such new operations was solved. Unfortunately, in the paper [1], the case where the initial groups contain involutions, was not analyzed in detail. It is shown that, in the case where the initial groups contain involutions, this small gap is easily removed by an additional restriction on the choice of defining relations for the periodic product. It suffices to simply exclude products of two involutions of previous ranks from the inductive process of defining new relations for any given rank α. It is suggested that the adequacy of the given restriction follows easily from the proof of the key Lemma II.5.21 in the monograph [2]. We also mention that, with this additional restriction, all the properties of the periodic product given in [1] remain true with obvious corrections of their formulation. Moreover, under this restriction, one can consider n-periodic products for any period n ≥ 665, including even periods.     

Free algebras of a unary variety with Mal’tsev’s operation that satisfies the Pixley conditions

In this paper we consider the variety V P of algebras with one unary and one ternary operation p that satisfies the Pixley identities, provided that operations are permutable. We describe the structure of a free algebra of the variety V P and study the structure of unary reducts of free algebras. We prove the solvability of the word problem in free algebras and the uniqueness of a free basis; we also describe groups of automorphisms of free algebras. Similar results are obtained for free algebras of a subvariety of the variety V P defined by the identities p(p(x, y, z), y, z) = p(x, y, z) and p(x, y, p(x, y, z)) = p(x, y, z).