Narrow Results

We prove that the Brinkmann Problems (BrP & BrCP) and the Twisted-Conjugacy Problem (TCP) are decidable for any endomorphism of a free-abelian times free (FATF) group ${𝔽}_n\times {\mathbb{Z}}^m$. Furthermore, we prove the decidability of the two-sided Brinkmann Conjugacy Problem (2BrCP) for monomorphisms of FATF groups (and combine it with TCP) to derive the decidability of the conjugacy problem for ascending HNN extensions of FATF groups.

We study the structure of quantum Markov Processes from the point of view of product systems and their representations.

In this paper, we first show that the monoid of separable surjective self-morphisms of a variety of Ueno type coincides with the group of automorphisms. We also give an explicit description of the automorphism group. As applications, we confirm Kawaguchi–Silverman conjecture for automorphisms of a variety of Ueno type and some Calabi–Yau three-fold, defined over $\overline{\mathbb{Q}}$.

The purpose of this paper is to study semigroups of *-endomorphisms of type I factors, with parameters in a countable dense subgroup of the real line. Most of our results are motivated by the theory of semigroups of isometries. It is shown that every semigroup of *-endomorphisms can be extended to a group of *-automorphisms in a minimal way. In certain situations, it is shown that a semigroup of *-endomorphisms can be decomposed into three simpler semigroups. Shifts are defined and classified up to outer conjugacy.

Invariant areas generated by two-dimensional endomorphisms are studied using the method of critical curves. The invariant areas considered in this paper are obtained by iterating the noninvariant set constituted by the connected basin of an attracting set or the immediate basin of a nonconnected basin. This new kind of invariant area is of mixed type in the sense that its boundary is made up of critical curves arcs and arcs of saddle manifolds. The presentation is illustrated by three examples. A bifurcation changing the degree of connexity of an invariant area is described.

Let G be the fundamental group of a graph of groups with finite edge groups and f an endomorphism of G. We prove a structure theorem for the subgroup Fix(f), which consists of the elements of G fixed by f, in the case where the endomorphism f of G maps vertex groups into conjugates of themselves.

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.

Infinite words over a finite special confluent rewriting system R are considered and endowed with natural algebraic and topological structures. Their geometric significance is explored in the context of Gromov hyperbolic spaces. Given an endomorphism φ of the monoid generated by R, existence and uniqueness of several types of extensions of φ to infinite words (endomorphism extensions, weak endomorphism extensions, continuous extensions) are discussed. Characterization theorems and positive decidability results are proved for most cases.

It is proved that the periodic point submonoid of a free inverse monoid endomorphism is always finitely generated. Using Chomsky's hierarchy of languages, we prove that the fixed point submonoid of an endomorphism of a free inverse monoid can be represented by a context-sensitive language but, in general, it cannot be represented by a context-free language.

The aim of the paper is to construct, discuss and apply the Galois-type correspondence between subsemigroups of the endomorphism semigroup $End(A)$ of an algebra $A$ and sets of logical formulas. Such Galois-type correspondence forms a natural frame for studying algebras by means of actions of different subsemigroups of $End(A)$ on definable sets over $A$. We treat some applications of this Galois correspondence. The first one concerns logic geometry. Namely, it gives a uniform approach to geometries defined by various fragments of the initial language. The next prospective application deals with effective recognition of sets and effective computations with properties that can be defined by formulas from a fragment of the original language. In this way, one can get an effective syntactical expression by semantic tools. Yet another advantage is a common approach to generalizations of the main model theoretic concepts to the sublanguages of the first-order language and revealing new connections between well-known concepts. The fourth application concerns the generalization of the unification theory, or more generally Term Rewriting Theory, to the logic unification theory.

The string number of self-maps arose in the context of algebraic entropy and it can be viewed as a kind of combinatorial entropy function. Later on, its values for endomorphisms of abelian groups were calculated in full generality. We study its global version for abelian groups, providing several examples involving also Hopfian abelian groups. Moreover, we characterize the class of all abelian groups with string number zero in many cases and discuss its stability properties.

In this paper, we characterize the monoid of endomorphisms of the semigroup of all monotone full transformations of a finite chain, as well as the monoids of endomorphisms of the semigroup of all monotone partial transformations and of the semigroup of all monotone partial permutations of a finite chain.

It is proved that the Jacobian of a k-endomorphism of k[x₁,…,x_n] over a field k of characteristic zero, taking every tame coordinate to a coordinate, must be a nonzero constant in k. It is also proved that the Jacobian of an R-endomorphism of A=:R[x₁,…,x_n] (where R is a polynomial ring in a finite number of variables over an infinite field k), taking every R-linear coordinate of A to an R-coordinate of A, is a nonzero constant in k.

Let G be a group. A G-set is a nonempty set A together with a (right) action of G on A. The class of G-sets, viewed as unary algebras, is a variety. For a set X, let A_G(X) be the free algebra on X in the variety of G-sets. We determine the group of automorphisms of End(A_G(X)), the monoid of endomorphisms of A_G(X).

This paper aims to investigate the commutativity of prime rings with involution * of the second kind in which endomorphisms satisfy some algebraic identities involving hermitian and skew-hermitian elements. We will also provide some classifications of these endomorphisms. Finally, we give some examples to prove that the imposed hypotheses are necessary.

We show that any unital normal *-endomorphism of a von Neumann algebra admits a Kallman type decomposition, i.e., it can be decomposed uniquely as a central direct sum of a family of k-inner endomorphisms and a properly outer endomorphism. This decomposition is stable under conjugacy and cocycle conjugacy.

We look at structure results in nearring theory, especially with regard to Jacobson style primitivity. We look at some applications in state automata and cellular automata, where we find interesting structures. We then look at generalising planarity and nearfields, some of the best structured nearrings, and find that Jacobson style primitivity plays an important role.

The paper explores some of our recent work and presents several ways forward that we think might be of value. While we hope to answer some of the questions that we raise here, it is likely that other researchers will find simpler ways forward that we are unable to see. We look forward to some surprises!