International Journal of Algebra and Computation

Let X be a connected partially ordered set and let K be a field of characteristic different from 2. We present necessary and sufficient conditions for two involutions on the finitary incidence algebra of X over K, FI(X), to be equivalent in the case when every multiplicative automorphism of FI(X) is inner. To get the classification of involutions we extend the concept of multiplicative automorphism to finitary incidence algebras and prove the Decomposition Theorem of involutions of [Anti-automorphisms and involutions on (finitary) incidence algebras, Linear Multilinear Algebra 60 (2012) 181–188] for finitary incidence algebras.

We prove that the class K(σ) of all algebraic structures of signature σ is Q-universal if and only if there is a class K ⊆ K(σ) such that the problem whether a finite lattice embeds into the lattice of K-quasivarieties is undecidable.

We study free subgroups in the multiplicative group of division rings generated by group rings of some classes of soluble groups.

A temporal constraint language is a set of relations that are first-order definable over (ℚ;<). We show that several temporal constraint languages whose constraint satisfaction problem is maximally tractable are also maximally tractable for the more expressive quantified constraint satisfaction problem. These constraint languages are defined in terms of preservation under certain binary polymorphisms. We also present syntactic characterizations of the relations in these languages.

Let K〈x_i〉 be the free associative algebra generated by a finite or a countable number of variables x_i. The notion of "letterplace correspondence" introduced in [R. La Scala and V. Levandovskyy, Letterplace ideals and non-commutative Gröbner bases, J. Symbolic Comput. 44(10) (2009) 1374–1393; R. La Scala and V. Levandovskyy, Skew polynomial rings, Gröbner bases and the letterplace embedding of the free associative algebra, J. Symbolic Comput. 48 (2013) 110–131] for the graded (two-sided) ideals of K〈x_i〉 is extended in this paper also to the nongraded case. This amounts to the possibility of modelizing nongraded noncommutative presented algebras by means of a class of graded commutative algebras that are invariant under the action of the monoid ℕ of natural numbers. For such purpose we develop the notion of saturation for the graded ideals of K〈x_i,t〉, where t is an extra variable and for their letterplace analogues in the commutative polynomial algebra K[x_ij, t_j], where j ranges in ℕ. In particular, one obtains an alternative algorithm for computing inhomogeneous noncommutative Gröbner bases using just homogeneous commutative polynomials. The feasibility of the proposed methods is shown by an experimental implementation developed in the computer algebra system Maple and by using standard routines for the Buchberger algorithm contained in Singular.

We present a number of new results on the multiplicative structure of univariate polynomials over the Boolean and tropical semirings. We answer the question asked by Kim and Roush in 2005 by proving that almost all Boolean polynomials with nonzero constant term are irreducible. We also give a lower bound for the number of reducible polynomials, and we discuss the related issues for polynomials over tropical semiring.

This work is about the structure of the symbolic Rees algebra of the base ideal of a Cremona map. We give sufficient conditions under which this algebra has the "expected form" in some sense. The main theorem in this regard seemingly covers all previous results on the subject so far. The proof relies heavily on a criterion of birationality and the use of the so-called inversion factor of a Cremona map. A pretty long selection of examples of plane and space Cremona maps has been given to test against the conditions of the theorem, with special emphasis on Cohen–Macaulay base ideals.

The conjugacy growth function counts the number of distinct conjugacy classes in a ball of radius n. We give a lower bound for the conjugacy growth of certain branch groups, among them the Grigorchuk group. This bound is a function of intermediate growth. We further prove that certain branch groups have the property that every element can be expressed as a product of uniformly boundedly many conjugates of the generators. We call this property bounded conjugacy width. We also show how bounded conjugacy width relates to other algebraic properties of groups and apply these results to study the palindromic width of some branch groups.

In this paper, we deal with a notion of Banach space-valued mappings defined on a set consisting of finite graphs with uniformly bounded vertex degree. These functions will be endowed with certain boundedness and additivity criteria. We examine their normalized long-term behavior along a particular class of graph sequences. Using techniques developed by Elek, we show convergence in the topology of the Banach space if the corresponding graph sequence possesses a hyperfinite structure. These considerations extend and complement the corresponding results for amenable groups. As an application, we verify the uniform approximation of the integrated density of states for bounded, finite range operators on discrete structures. Further, we extend results concerning an abstract version of Fekete's lemma for cancellative, amenable groups and semigroups to the geometric situation of convergent graph sequences.