Narrow Results

We show that certain finite groups do not arise as the automorphism group of the square of a finite algebraic structure, nor as the automorphism group of a finite, 2-generated, free, algebraic structure.

Let U be a universal algebra. An automorphism α of the endomorphism semigroup of U defined by α(φ) = sφs^-1 for a bijection s : U → U is called a quasi-inner automorphism. We characterize bijections on U defining such automorphisms. For this purpose, we introduce the notion of a pre-automorphism of U. In the case when U is a free universal algebra, the pre-automorphisms are precisely the well-known weak automorphisms of U. We also provide different characterizations of quasi-inner automorphisms of endomorphism semigroups of free universal algebras and reveal their structure. We apply obtained results for describing the structure of groups of automorphisms of categories of free universal algebras, isomorphisms between semigroups of endomorphisms of free universal algebras, automorphism groups of endomorphism semigroups of free Lie algebras etc.

Modes are idempotent and entropic algebras. Although it had been established many years ago that groupoid modes embed as subreducts of semimodules over commutative semirings, the general embeddability question remained open until Stronkowski and Stanovský's recent constructions of isolated examples of modes without such an embedding. The current paper now presents a broad class of modes that are not embeddable into semimodules, including structural investigations and an analysis of the lattice of varieties.

Differential modes provide examples of modes that do not embed as subreducts into semimodules over commutative semirings. The current paper studies differential modes, so-called Szendrei differential modes, which actually do embed into semimodules. These algebras form a variety. The main result states that the lattice of nontrivial subvarieties is dually isomorphic to the (nonmodular) lattice of congruences of the free commutative monoid on two generators. Consequently, all varieties of Szendrei differential modes are finitely based.

Let $P_n=k[x_1,x_2,\dots ,x_n]$ be the polynomial algebra over a field $k$ of characteristic zero in the variables $x_1,x_2,\dots ,x_n$ and ${\mathcal{ℒ}}_n$ be the left-symmetric Witt algebra of all derivations of $P_n$ [D. Burde, Left-symmetric algebras, or pre-Lie algebras in geometry and physics, Cent. Eur. J. Math.4(3) (2006) 323–357]. We describe all right operator identities of ${\mathcal{ℒ}}_n$ and prove that the set of all algebras ${\mathcal{ℒ}}_n$, where $n\ge 1$, generates the variety of all left-symmetric algebras. We also describe a class of general (not only right operator) identities for ${\mathcal{ℒ}}_n$.

A variety of associative algebras over a field of characteristic 0 is called minimal if the exponent of the variety which measures the growth of its codimension sequence is strictly larger than the exponent of any of its proper subvarieties, i.e., its codimension sequence grows much faster than the codimension sequence of its proper subvarieties. By the results of Giambruno and Zaicev it follows that the number $b_n$ of minimal varieties of given exponent $n$ is finite. Using methods of the theory of colored (or weighted) compositions of integers, we show that the limit $\beta ={lim}_{n\to \infty }\sqrt[n]{b_n}$ exists and can be expressed as the positive solution of an equation $a(t)=0$ where $a(t)$ is an explicitly given power series. Similar results are obtained for the number of minimal varieties with a given Gelfand–Kirillov dimension of their relatively free algebras of rank $d$. It follows from classical results on lacunary power series that the generating function of the sequence $b_n$, $n=1,2,\dots$, is transcendental. With the same approach we construct examples of free graded semigroups $〈Y〉$ with the following property. If $d_n$ is the number of elements of degree $n$ of $〈Y〉$, then the limit $\delta ={lim}_{n\to \infty }\sqrt[n]{d_n}$ exists and is transcendental.

Define an augmented LD-system, or ALD-system, to be a set equipped with two binary operations, one satisfying the left self-distributivity law x * (y * z) = (x * y) * (x * z) and the other satisfying the mixed laws (x ◦ y) * z = x * (y * z) and x * (y ◦ z) = (x * y) ◦ (x * z). We solve the word problem of the ALD laws, and prove that every element in the parenthesized braid group B_• of [2, 3, 5, 6] generates a free ALD-system of rank 1, i.e., that B_• is a torsion-free ALD-system.

D. Puder defined the primitivity rank of elements of free groups [Primitive words, free factors and measure preservation, Israel J. Math.201(1) (2014) 25–73], we give a similar definition for free algebras of Schreier varieties and prove properties of a primitivity rank using the properties of the almost primitive elements.