Narrow Results

The algebraic composition of an element $x$ of a commutative binary structure ${ℬ}$ is modeled by its divisor (pseudo)graph ${\mathrm{\Gamma }}_x$ (which can be regarded as a simple graph if ${ℬ}$ is an abelian group), whose vertices are the elements of ${ℬ}$ such that two vertices $a$ and $b$ are adjacent if and only if $ab=x$. With respect to the action of a group of symmetries on the set of divisor pseudographs of ${ℬ}$, the numbers of orbits are considered, and these numbers are shown to yield groupoid-isotopy invariants when ${ℬ}$ is a finite abelian group. The minimum number of orbits is computed for every abelian group of order at most 11, and it is also determined for all finite cyclic groups. Moreover, systems of pseudographs that can be realized as those of finite abelian groups are completely characterized. In fact, recurrence relations are given for constructing systems of divisor pseudographs of finite cyclic groups, and all commutative binary structures that are isotopic to a finite abelian group are established.

Let ${{𝒜}}_0$ be the class of Abelian block-rigid almost completely decomposable groups of ring type with cyclic regulator quotient. In this paper, we study relationships between the above groups $G$ and their multiplication groups Mult $G$. It is proved that groups from ${{𝒜}}_0$ are determined by their multiplication groups. For a rigid group $G\in {{𝒜}}_0$, the isomorphism problem is solved: we describe multiplications from $\text{Mult}G$ that define isomorphic rings on $G$. We also describe Abelian groups that are realized as the multiplication group of some group in ${{𝒜}}_0$. We describe groups in ${{𝒜}}_0$ that are isomorphic to their multiplication groups.

In this paper we prove that the pseudovariety of Abelian groups is hyperdecidable and moreover that it is completely tame. This is a consequence of the fact that a system of group equations on a free Abelian group with certain rational constraints is solvable if and only if it is solvable in every finite quotient.

Hedrlín and Pultr proved that for any monoid M there exists a graph G with endomorphism monoid isomorphic to M. In a previous paper, we give a construction G(M) for a graph with prescribed endomorphism monoid M known as a -graph. Using this construction, we derived bounds on the minimum number of vertices and edges required to produce a graph with a given endomorphism monoid for various classes of finite monoids. In this paper, we generalize the -graph construction and derive several new bounds for monoid classes not handled by our first paper. Among these are the so called "strong semilattices of C-semigroups" where C is one of the following: Groups, Abelian Groups, Rectangular Groups, and completely simple semigroups.

Abelian groups are classified by the existence of certain additive decompositions of group-valued functions of several variables.

We use computability-theoretic tools to measure the complexity of the process of direct decomposing an abelian group based on its symbolic presentation. More specifically, we compare degrees of decidable categoricity of abelian groups with degrees of categoricity of their natural direct summands. As a nontrivial and unexpected application of our methods, we show that every decidable copy of a nondivisible homogeneous completely decomposable group has an algorithm for linear independence.

We comment on the paper “Extremal Cayley digraphs of finite Abelian groups” [Intercon. Networks 12 (2011), no. 1-2, 125–135]. In particular, we give some counterexamples to the results presented there, and provide a correct result for degree two.

An R-module _RM is called morphic if M/im α ≅ ker α for every endomorphism α of M, that is, if the dual of the Noether isomorphism theorem holds. Mostly all morphic Z-modules are determined leaving open some classes of nonsplitting mixed groups, which actually cannot be completely characterized. However, for these classes, comprehensive information including negative examples is given.

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 will extend the notion of zero-divisor graph of commutative rings to zero-divisor graph of abelian groups and study this zero-divisor graph. We characterize the zero-divisor graph of almost all abelian groups. Just a few classes of reduced abelian groups remain untouched in this paper.

We introduce the notion of intrinsic semilattice entropy $\tilde{h}$ in the category ${{ℒ}}_{\mathrm{\text{qm}}}$ of generalized quasimetric semilattices and contractive homomorphisms. By using appropriate categories $𝔛$ and functors $F:𝔛\to {{ℒ}}_{\mathrm{\text{qm}}}$, we find specific known entropies ${\tilde{h}}_{𝔛}$ on $𝔛$ as intrinsic functorial entropies, that is, as ${\tilde{h}}_{𝔛}=\tilde{h}\circ F$. These entropies are the intrinsic algebraic entropy, the algebraic and the topological entropies for locally linearly compact vector spaces, the topological entropy for totally disconnected locally compact groups and the algebraic entropy for compactly covered locally compact abelian groups.

The paper contains the classification of non-commutative torsion-free associative rings of rank two. Furthermore, torsion-free abelian groups of rank two supporting an associative, but not commutative ring structure are classified.

A multiplication on an Abelian group $G$ is a homomorphism $\mu :G\otimes G\to G$. The set $\mathrm{\text{Mult}}G$ of all multiplications on an Abelian group $G$ itself is an Abelian group with respect to addition; this group is called the multiplication group of $G$. In the paper, the class $\mathcal{𝒬}\mathcal{𝒟}1$ of quotient divisible Abelian groups of rank 1 is considered. The group $\mathrm{\text{Mult}}G$ is described for $G\in \mathcal{𝒬}\mathcal{𝒟}1$. In the group $\mathrm{\text{Mult}}G$, multiplications defining comparable rings on $G$ are described for any group $G\in \mathcal{𝒬}\mathcal{𝒟}1$. The isomorphism problem is solved in $\mathcal{𝒬}\mathcal{𝒟}1$: multiplications defining isomorphic rings on $G$ are described for any $G\in \mathcal{𝒬}\mathcal{𝒟}1$.

The intersection graph of subgroups of a finite group $G$ is a graph whose vertices are all nontrivial subgroups of $G$ and in which two distinct vertices $H$ and $K$ are adjacent if and only if $H\cap K\ne 1$. The non-orientable genus of a graph $\mathrm{\Gamma }$ is the smallest positive integer $n$ such that $\mathrm{\Gamma }$ can be embedded on ${𝕊}_n$ (${\mathbb{N}}_n$), where ${𝕊}_n$ and ${\mathbb{N}}_n$ are the surface obtained from the sphere by attaching $n$ handles and the sphere with $n$ added crosscaps, respectively. In this paper, we classify all finite abelian groups whose non-oritentable genus of intersection graphs of subgroups are 1–3, respectively.

It is shown that the Baer–Kaplansky Theorem can be extended to all abelian groups provided that the rings of endomorphisms of groups are replaced by trusses of endomorphisms of corresponding heaps. That is, every abelian group is determined up to isomorphism by its endomorphism truss and every isomorphism between two endomorphism trusses associated to some abelian groups $G$ and $H$ is induced by an isomorphism between $G$ and $H$ and an element from $H$. This correspondence is then extended to all modules over a ring by considering heaps of modules. It is proved that the truss of endomorphisms of a heap associated to a module $M$ determines $M$ as a module over its endomorphism ring.

In [M. Chandramouleeswaran and P. Muralikrishna, The recipotent abelian group, Int. J. Math. Sci.5(4) (2010) 191–200], the concept of a recipotent matrix was introduced. A matrix $A$ of order $n$ is called recipotent matrix if $A\circ A^R=0_n$ where $A^R$ is reversible matrix of $A$, defined by each nonzero element of $A$ is replaced by its reciprocal. The aim of this paper is to improve and complete that paper.

In this paper we consider the problem of computation of a basis for an abelian group G with N elements such that the prime factorization of N is known. We present two deterministic algorithms for this task and a deterministic algorithm in case where a generating system for G is given.