In this work we study the self-similarity of infinite products of the group of the integers. In [A. C. Dantas, T. M. G. Santos and S. N. Sidki, Self-similar abelian groups and their centralizers, Groups Geom. Dyn. 17 (2023) 1–23], the authors ask if there is a self-similar free abelian group of uncountable rank. On the other hand, in [L. Bartholdi and S. N. Sidki, Self-similar products of groups, Groups Geom. Dyn. 14(1) (2020) 107–115] is asked if the Baer–Specker group [Formula: see text] is self-similar (by a result of R. Baer, this group is not free abelian). We answer both questions positively.

Abstract Assuming Gödel's axiom of constructibility , we construct a ‐free abelian group of singular cardinality for some suitable cardinal which is regular and uncountable, equipped with the property that for every nontrivial subgroup of smaller cardinality, , while . This provides a consistent counterexample to the singular compactness of nontrivial duality with respect to the functor .

We construct the free group over a non-Archimedean fuzzy metric space (X,M,∧) in the sense of George and Veeramani where ∧ is the minimum t-norm. The two main tools used are the concept of a scheme (for every non-empty subset S of N of even cardinal, a permutation φ on S is a scheme for S if it is idempotent, with no fixed points and, additionally, i<j<φ(i)<φ(j) does not hold for every i,j∈S), and the notion of a fuzzy prenorm on a fuzzy topological group. As a consequence of our results, we prove that every non-Archimedean fuzzy metric space (X,M,∧) in the sense of George and Veeramani is isometric to a closed subspace of a non-Archimedean fuzzy metric free (Abelian) group and also that every metric space (X,d) is uniformly isomorphic to a closed subspace of a non-Archimedean fuzzy metric free (Abelian) group. Our results also apply to non-Archimedean fuzzy metric spaces in the sense of Kramosil and Michálek.

Let [Formula: see text] be a group, and let [Formula: see text] be an automorphism of [Formula: see text]. If [Formula: see text] then [Formula: see text] is said to be a commuting automorphism. The set of all such automorphisms is denoted by [Formula: see text]. This set does not necessarily form a subgroup of the automorphism group of [Formula: see text]. If [Formula: see text] does form a subgroup, then [Formula: see text] is said to be an [Formula: see text]-group. Let [Formula: see text] be a set of prime numbers. Define [Formula: see text] as the ring consisting of all rational numbers [Formula: see text], where [Formula: see text] and [Formula: see text] are coprime integers, and [Formula: see text] is a [Formula: see text]-number. The additive group of [Formula: see text] is denoted by [Formula: see text]. Now let [Formula: see text] and [Formula: see text] be two sets of primes, and let [Formula: see text] be a nonzero integer. Consider a generalized extraspecial [Formula: see text]-group [Formula: see text], defined as follows: [Formula: see text] Let [Formula: see text], where [Formula: see text] is a generalized extraspecial [Formula: see text]-group such that [Formula: see text] with [Formula: see text]. In this paper, we show that if [Formula: see text], then [Formula: see text] is a non-[Formula: see text]-group, and if [Formula: see text], then [Formula: see text] is an [Formula: see text]-group. As a consequence, we identify the conditions determining when the following groups are [Formula: see text]-groups or not: (i) the direct product of a generalized extraspecial [Formula: see text]-group and a free abelian group with finite rank, (ii) an extension of [Formula: see text] by a direct sum of finitely many copies of [Formula: see text], where [Formula: see text] is the additive group of rational numbers, (iii) an infinite Černikov [Formula: see text]-group which is non-abelian but each proper quotient group is abelian.

ABSTRACT In this work, we introduce a notion of tensor product of (twisted) quiver representations with relations in the category of $\mathcal {O}_X$-modules. As a first application of our notion, we see that tensor products of polystable quiver bundles are polystable and later we use this to both deduce a quiver version of the Segre embedding and to identify distinguished closed subschemes of $\text{GL}(n,\mathbb {C})$-character varieties of free abelian groups.

Abstract Let S be a fine and saturated (fs) log scheme, and let F be a group scheme over the underlying scheme of S which is étale locally representable by (1) a finite dimensional $\mathbb{Q}$ -vector space, or (2) a finite rank free abelian group, or (3) a finite abelian group. We give a full description of all the higher direct images of F from the Kummer log flat site to the classical flat site. In particular, we show that: in case (1) the higher direct images of F vanish; and in case (2) the first higher direct image of F vanishes and the nth ( $n\gt 1$ ) higher direct image of F is isomorphic to the $(n-1)$ -th higher direct image of $F\otimes_{{\mathbb Z}}{\mathbb Q}/{\mathbb Z}$ . In the end, we make some computations when the base is a standard henselian log trait or a Dedekind scheme endowed with the log structure associated to a finite set of closed points.

In this paper, we investigate the finiteness properties of subgroups of direct products of 2-dimensional coherent right-angled Artin groups. We explore how these properties relate to the structure of the subgroups and the decidability of certain algorithmic problems. More precisely, we show that a finitely presented subgroup S of the direct product of 2-dimensional coherent RAAGs is virtually a nilpotent extension of a direct product. Moreover, if S is of type FP, then S is commensurable to a kernel of a character. We use these results to show that the multiple conjugacy problem and the membership problem are decidable for finitely presented subgroups of direct products of 2-dimensional coherent RAAGs. This work generalizes the results of Bridson, Howie, Miller, and Short for free groups.

Abstract Let be a submonoid of a free Abelian group of finite rank. We show that if is a field of prime characteristic such that the monoid ‐algebra is , then is a finitely generated ‐algebra, or equivalently, that is a finitely generated monoid. Split‐‐regular rings are possibly non‐Noetherian or non‐‐finite rings that satisfy the defining property of strongly ‐regular rings from the theories of tight closure and ‐singularities. Our finite generation result provides evidence in favor of the conjecture that rings in function fields over have to be Noetherian. The key tool is Diophantine approximation from convex geometry.

It is proved that the classes of torsion free abelian groups and divisible torsion free abelian groups have unique positive existentially closed model up to isomorphism. we give complete description of h-amalgamation bases in the class of non-trivial torsion free abelian groups.

Magnetic vortices and skyrmions are typically characterized by distinct topological invariants. This paper presents a unified approach for the topological classification of these textures, encompassing isolated objects and configurations where skyrmions and vortices coexist. Using homotopy group analysis, we derive topological invariants that form the free Abelian group, Z×Z. We provide an explicit method for calculating the corresponding integer indices in continuous and discrete systems. This unified classification framework extends beyond magnetism and is applicable to physical systems in general. Published by the American Physical Society 2025

Abstract This paper opens and discusses the question originally due to Daniel Herden, who asked for which graph ( μ , R ) {(\mu,R)} we can find a family { 𝔾 α : α < μ } {\{\mathbb{G}_{\alpha}:\alpha<\mu\}} of abelian groups such that for each α , β ∈ μ {\alpha,\beta\in\mu} , Ext ⁢ ( 𝔾 α , 𝔾 β ) = 0 {{\rm Ext}(\mathbb{G}_{\alpha},\mathbb{G}_{\beta})=0} iff ( α , β ) ∈ R {(\alpha,\beta)\in R} . In this regard, we present four results. First, we give a connection to Quillen’s small object argument which helps Ext {{\rm Ext}} vanishes and use it to present a useful criteria to the question. Suppose λ = λ ℵ 0 {\lambda=\lambda^{\aleph_{0}}} and μ = 2 λ {\mu=2^{\lambda}} . We apply Jensen’s diamond principle along with the criteria to present λ-free abelian groups representing bipartite graphs. Third, we use a version of the black box to construct in ZFC, a family of ℵ 1 {\aleph_{1}} -free abelian groups representing bipartite graphs. Finally, applying forcing techniques, we present a consistent positive answer for general graphs.

In this paper, we study (logical) types and isotypical equivalence of torsion-free Abelian groups. We describe all possible types of elements and standard 2-tuples of elements in these groups, and we classify separable torsion-free Abelian groups up to isotypicity.

I have developed an algorithm and written a GAP functions pautfreegroup:=function(n) (Finite Presentation of automorphism Groups of Free group of Rank n), and autfreeabeliangroup:=function(n) (Finite Presentation of automorphism Groups of Free abelian group of Rank n). Functions using GAP system for computation of a finite presentation for Aut(F_n ) and GL(n,z) respectively. In order to do that we have given a description for the presentation of Aut(F_n) and GL(n,z), the automorphism groups of free groups and free abelian groups respectively.

The geometry of a flag complex is explored and the homology group $H_1$ computed for the complete flag $F(s)$ by viewing it as a simplicial complex. The homomorphisms between the vertices are extended to the free abelian groups generated by the vertices and it is proved that $H_1(F(s))\cong\underbrace{\mathbb Z+\mathbb Z+\cdots+\mathbb Z}_{k \ \hbox{times}}, \ k\geq1$ where $k=s-r$ , $r$ the rank of a matrix $A$ associated to $F(s)$.

Abstract Let X be a smooth projective variety of dimension $n\geq 2$ and $G\cong \mathbf {Z}^{n-1}$ a free abelian group of automorphisms of X over $\overline {\mathbf {Q}}$ . Suppose that G is of positive entropy. We construct a canonical height function $\widehat {h}_G$ associated with G, corresponding to a nef and big $\mathbf {R}$ -divisor, satisfying the Northcott property. By characterizing the zero locus of $\widehat {h}_G$ , we prove the Kawaguchi–Silverman conjecture for each element of G. As for other applications, we determine the height counting function for non-periodic points and show that X satisfies potential density.

Given a cyclically ordered free Abelian group G, a unital C^*-algebra A, and a semigroup homomorphism alpha from P(G) into the semigroup Endo(A), we characterize faithful representations of crossed product Ax_{alpha} P(G) by endomorphisms.

Abstract The group of order-preserving automorphisms of a finitely generated Archimedean ordered group of rank $2$ is either infinite cyclic or trivial according as the ratio in $\mathbb {R}$ of the generators of the subgroup is or is not quadratic over $\mathbb {Q}.$ In the case of an Archimedean ordered group of rank $2$ that is not finitely generated, the group of order-preserving automorphisms is free abelian. Criteria determining the rank of this free abelian group are established.

A Bieberbach group is a torsion free crystallographic group that represents an extension of a free abelian lattice group by a finite point group. This research began by taking the group offered in the Crystallographic Algorithms and Tables (CARAT) package, which is in the matrix form. There are only four Bieberbach groups of dimension six to be isomorphic to the quaternion point group of order eight. In this study, three Bieberbach groups of dimension six with the quaternion point group of order eight that are considered as only the first group has been found its well-defined polycyclic presentation. Every group has eight generators that describe the group. However, the algorithm used in constructing the polycyclic presentation requires a new arbitrary generator to be added into the group. Then the consistency relations need to be checked and the polycyclic presentation is said to be a well-defined construction if it is consistent. Therefore, this study shows the construction of polycyclic presentation with the new arbitrary generator for all three groups. Furthermore, the polycyclic presentation for the second group has been proven to be consistent, which implies that the construction is well-defined.

We prove that the set of subgroups of the automorphism group of a two-sided full shift is closed under countable graph products. We introduce the notion of a group action without [Formula: see text]-cancellation (for an abelian group [Formula: see text]), and show that when [Formula: see text] is a finite abelian group and [Formula: see text] is a group of cellular automata whose action does not have [Formula: see text]-cancellation, the wreath product [Formula: see text] embeds in the automorphism group of a full shift. We show that all free abelian groups and free groups admit such cellular automata actions. In the one-sided case, we prove variants of these results with reasonable alphabet blow-ups.

Abstract A ring $$\Lambda $$ Λ has stably free cancellation when every stably free $$\Lambda $$ Λ -module is free. Let $$ G \; = \; C_p \rtimes C_q $$ G = C p ⋊ C q be a finite metacyclic group where p is an odd prime and q is a positive integral divisor of $$p-1$$ p - 1 . We show that the group ring $$\mathcal{R}[G]$$ R [ G ] has stably free cancellation when $$\;\mathcal{R} \; = \; {\mathbb {Z}}[t_1, t_1^{-1}, \dots t_m, t_m^{-1}, x_1, \dots x_n] \;$$ R = Z [ t 1 , t 1 - 1 , ⋯ t m , t m - 1 , x 1 , ⋯ x n ] is a ring of mixed polynomials and Laurent polynomials over the integers. As a consequence, when $$C_\infty ^{(m)}$$ C ∞ ( m ) is the free abelian group of rank m then the integral group ring $${\mathbb {Z}}[G(p,q) \times C_\infty ^{(m)}]\; $$ Z [ G ( p , q ) × C ∞ ( m ) ] has stably free cancellation.