The Cuboid Lemma and Mal’tsev Categories

We prove that a regular category ? is a Mal’tsev category if and only if a strong form of the denormalised 3 × 3 Lemma holds true in ?. In this version of the 3 × 3 Lemma, the vertical exact forks are replaced by pullbacks of regular epimorphisms along arbitrary morphisms. The shape of the diagram it determines suggests to call it the Cuboid Lemma. This new characterisation of regular categories that are Mal’tsev categories (= 2-permutable) is similar to the one previously obtained for Goursat categories (= 3-permutable). We also analyse the “relative” version of the Cuboid Lemma and extend our results to that context.     

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.     

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).     

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.     

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.     

2-Supernilpotent Mal’cev algebras

In this note we prove that a Mal’cev algebra is 2-supernilpotent ([1, 1, 1] =  0) if and only if it is polynomially equivalent to a special expanded group. This generalizes Gumm’s result that a Mal’cev algebra is abelian if and only if it is polynomially equivalent to a module over a ring.     

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.     

Some applications of higher commutators in Mal’cev algebras

We establish several properties of Bulatov’s higher commutator operations in congruence permutable varieties. We use higher commutators to prove that for a finite nilpotent algebra of finite type that is a product of algebras of prime power order and generates a congruence modular variety, affine completeness is a decidable property. Moreover, we show that in such algebras, we can check in polynomial time whether two given polynomial terms induce the same function.     

Polynomial clones of Mal’cev algebras with small congruence lattices

We show that the polynomials of every finite Mal’cev algebra with congruence lattice of height at most 2 can be described by a finite set of relations.     

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.     

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.     

Uniform Mal’cev algebras with small congruence lattices

A congruence of an algebra is called uniform if all the congruence classes are of the same size. An algebra is called uniform if each of its congruences is uniform. All algebras with a group reduct have this property. We prove that almost every finite uniform Mal’cev algebra with a congruence lattice of height at most two is polynomially equivalent to an expanded group.