Cev Algebras Research Articles

On polynomial completeness properties of finite Mal’cev algebras

Polynomial completeness results aim at characterizing those functions that are induced by polynomials. Each polynomial function is congruence preserving, but the opposite need not be true. A finite algebraic structure [Formula: see text] is called strictly 1-affine complete if every unary partial function from a subset of [Formula: see text] to [Formula: see text] that preserves the congruences of [Formula: see text] can be interpolated by a polynomial function of [Formula: see text]. The problem of characterizing strictly 1-affine complete finite Mal’cev algebras is still open. In this paper, we extend the characterization by Aichinger and Idziak of strictly 1-affine complete expanded groups to finite congruence regular Mal’cev algebras.

A sum-bracket theorem for simple Lie algebras

Let g be an algebra over K with a bilinear operation [⋅,⋅]:g×g→g not necessarily associative. For A⊆g, let Ak be the set of elements of g written combining k elements of A via + and [⋅,⋅].We show a “sum-bracket theorem” for simple Lie algebras over K of the form g=sln,son,sp2n,e6,e7,e8,f4,g2: if char(K) is not too small, we have growth of the form |Ak|≥|A|1+ε for all generating symmetric sets A away from subfields of K. Over Fp in particular, we have a diameter bound matching the best analogous bounds for groups of Lie type [2].As an independent intermediate result, we prove also an estimate of the form |A∩V|≤|Ak|dim⁡(V)/dim⁡(g) for linear affine subspaces V of g. This estimate is valid for all simple algebras, and k is especially small for a large class of them including associative, Lie, and Mal'cev algebras, and Lie superalgebras.

On the number of universal algebraic geometries

The algebraic geometry of a universal algebra {textbf{A}} is defined as the collection of solution sets of systems of term equations. Two algebras {textbf{A}}_1 and {textbf{A}}_2 are called algebraically equivalent if they have the same algebraic geometry. We prove that on a finite set A with |A| there are countably many algebraically inequivalent Mal’cev algebras and that on a finite set A with |A| there are continuously many algebraically inequivalent algebras.

On supernilpotent algebras

We establish a characterization of supernilpotent Mal’cev algebras which generalizes the affine structure of abelian Mal’cev algebras and the recent characterization of 2-supernilpotent Mal’cev algebras. We then show that for varieties in which the two-generated free algebra is finite: (1) neutrality of the higher commutators is equivalent to congruence meet-semidistributivity, and (2) the class of varieties which interpret a Mal’cev term in every supernilpotent algebra is equivalent to the existence of a weak difference term. We then establish properties of the higher commutator in the aforementioned second class of varieties.

Finiteness properties of direct products of algebraic structures

We consider the preservation of properties of being finitely generated, being finitely presented and being residually finite under direct products in the context of different types of algebraic structures. The structures considered include Mal'cev algebras (including groups, rings and other classical algebras, as well as loops), idempotent algebras (including lattices), semigroups, and algebras in congruence modular varieties. We aim to identify as broad classes as possible in which the ‘expected’ preservation results (A×B satisfies property P if and only if A and B satisfy P) hold, and to exhibit ways in which they may fail outside those classes.

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.

SUPERNILPOTENCE PREVENTS DUALIZABILITY

AbstractWe address the question of the dualizability of nilpotent Mal’cev algebras, showing that nilpotent finite Mal’cev algebras with a nonabelian supernilpotent congruence are inherently nondualizable. In particular, finite nilpotent nonabelian Mal’cev algebras of finite type are nondualizable if they are direct products of algebras of prime power order. We show that these results cannot be generalized to nilpotent algebras by giving an example of a group expansion of infinite type that is nilpotent and nonabelian, but dualizable. To our knowledge this is the first construction of a nonabelian nilpotent dualizable algebra. It has the curious property that all its nonabelian finitary reducts with group operation are nondualizable. We were able to prove dualizability by utilizing a new clone theoretic approach developed by Davey, Pitkethly, and Willard. Our results suggest that supernilpotence plays an important role in characterizing dualizability among Mal’cev algebras.

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.

THE SUBPOWER MEMBERSHIP PROBLEM FOR MAL'CEV ALGEBRAS

Given tuples a1, …, ak and b in An for some algebraic structure A, the subpower membership problem asks whether b is in the subalgebra of An that is generated by a1, …, ak. For A a finite group, there is a folklore algorithm which decides this problem in time polynomial in n and k. We show that the subpower membership problem for any finite Mal'cev algebra is in NP and give a polynomial time algorithm for any finite Mal'cev algebra with finite signature and prime power size that has a nilpotent reduct. In particular, this yields a polynomial algorithm for finite rings, vector spaces, algebras over fields, Lie rings and for nilpotent loops of prime power order.

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.

Polynomial functions on subdirect products

A congruence preserving function on a subdirect product of two finite Mal’cev algebras is polynomial if it induces polynomial functions on the subdirect factors and there are no skew congruences between the projection kernels. As a special case, if the direct product A × B of finite algebras A and B in a congruence permutable variety has no skew congruences, then the polynomial functions on A × B are exactly direct products of polynomials on A and on B. These descriptions apply in particular to classical polynomial functions on nonassociative rings. Also, for finite algebras A, B in a variety with majority term, the polynomial functions on A × B are exactly the direct products of polynomials on A and on B. However in arbitrary congruence distributive varieties the corresponding result is not true.

On identities of free finitely generated alternative algebras over a field of characteristic 3

In 1981 Filippov solved in the affirmative Shestakov's problem on the strictness of the inclusions in the chains of varieties generated by free alternative and Mal'cev algebras of finite rank over a field of characteristic distinct from 2 and 3. In the present paper an analogous result is proved for alternative algebras over a field of characteristic 3. The proof is based on the construction of three families of identities that hold on the algebras of the corresponding rank. A disproof of the identities on algebras of larger rank is carried out with the help of a prime commutative alternative algebra. It is also proved that in varieties of alternative algebras of finite basis rank over a field of characteristic 3 every soluble algebra is nilpotent.

Continuity of derivations on Mal'cev H*-algebras

A long time ago the concept of H*-algebra was introduced by Ambrose in [1] where the structure of complex associative H*-algebras was given. Since then this theory was extended to such classical types of non-associative algebras as alternative algebras (in [6]), Jordan algebras (in [5, 13, 14]), non-commutative Jordan algebras (in [5]), Lie algebras (in [3, 9, 10]) and Mal'cev algebras (in [2]).

The ‘closed subgroup theorem’ for localic herds and pregroupoids

In this paper we extend the result that a localic subgroup is necessarily closed, and the ‘fibrewise’ generalization of this result to localic groupoids, to algebras in the category of locales for more general algebraic theories such as quasigroups and herds (associative Mal'cev algebras), and to pregroupoids in the sense of A. Kock.

STRUCTURE AND REPRESENTATION OF BINARY-LIE ALGEBRAS

In this paper semisimple finite-dimensional binary-Lie algebras over a field of characteristic 0 are described. It is proved that a finite-dimensional binary-Lie semisimple algebra over a field of characteristic 0 is a Mal'cev algebra, and an arbitrary finite-dimensional irreducible binary-Lie module over a semisimple Mal'cev algebra is Mal'cev. Bibliography: 10 titles.

Mal'cev algebras satisfying Engel's condition

Mal'cev algebras satisfying Engel's condition

Characteristically Nilpotent Algebras

Our aim in this paper is to extend (Theorem 1.7) to general algebras a classical result of Lie algebras due to Léger and Togo [6]. This extension requires, in turn, extension to general algebras of the concept of characteristically nilpotent algebras introduced by Dixmier and Lister [3] for Lie algebras. Based on this extended concept, we introduce in § 2 a new concept of radical (and semisimplicity) for general algebras and Lie triple systems. We study in some detail the consequences of the newly introduced concepts, furnishing necessary examples. With a stronger notion of characteristically nilpotent Mal'cev algebra arising out of these concepts, we obtain (Proposition 3.6) for such an algebra the parallel to the Leger-Tôgô result mentioned at the outset. In § 4, we deal with a further generalization of the concept of characteristic nilpotency leading to extension of very recent results of Chao [1] and Tôgô [12].