Orderconvergence provides a unifying framework for the study of point convergence, boundedness, and filter merotopy, and plays a central role in both Convenient Topology and Bounded Topology. It offers a flexible approach to convergence phenomena that extends beyond the limitations of purely topological methods. In this paper, we investigate spaces equipped with order convergence from a categorical point of view and analyze the structural properties of the categories naturally associated with them.

It is well known that several classical categories arising in topology and convergence theory fail to satisfy desirable convenience properties, such as being Cartesian closed, extensional, or stable under the formation of quotients and products. In particular, the category TOP lacks these structural features, while the category CHY of Cauchy spaces, although Cartesian closed, does not form a quasitopos. Moreover, the full subcategory EF-PROX of CHY, consisting of Efremovič proximity spaces, also fails to possess the expected convenient properties.

To address these shortcomings, we introduce and study an appropriate supercategory of order convergence spaces together with suitable morphisms. We show that this category forms a strong topological universe in the sense of Convenient Topology, and in particular a quasitopos. As a consequence, quotients in this setting are stable under arbitrary products. This categorical framework enables a uniform treatment of bornological and Cauchy structures and provides a clearer understanding of their mutual relationships and categorical behavior. To exclude pathological cases, all underlying sets are assumed to be non-empty.

In this joint work with Pawel Gladki (University of Silesia, Poland) and Kaique M.A. Roberto (Faculty Einstein, Brazil), we develop a new approach to local–global principles in (non-reduced) algebraic theory of quadratic forms by employing instruments and techniques of hyper compositional algebra (a.k.a. multi-algebras).

We define quadratic extensions of special hyperfields (hyperfields that faithfully encode quadratic forms theory) as a particular type of Marshall quotients and investigate their basic properties. In particular we apply these constructions to develop new types of local–global principles in the algebraic theory of quadratic forms that we call n-th Pfister approximation properties (n-PAP) and analyze a few instances where they do—as well as do not—hold. This approach builds on Pfister's and Marshall's local–global principles that have been established in the realm of reduced theories of quadratic forms.

References:
[1] Pawel Gladki, Krzysztof Worytkiewicz. "Witt rings of quadratically presentable fields". Categories and General Algebraic Structures 12(1): 1-23, 2020.
[2] Hugo R. O. Ribeiro, Kaique M. A. Roberto, Hugo L. Mariano. "Functorial relationship between multirings and the various abstract theories of quadratic forms". São Paulo Journal of Mathematical Sciences 16(1):5–42, 2022.
[3] Kaique M. A. Roberto, Hugo R. O. Ribeiro, Hugo L. Mariano. "Quadratic structures associated to (multi) rings". Categories and General Algebraic Structures 16(1):105–141, 2022.

Let (Bn)n≥0 be the sequence of Balancing numbers defined by the recurrence relation Bn+2 = 6Bn+1 − Bn, n ≥ 0, with initial terms B0 = 0 and B1 = 1. In this work, we determine all Balancing numbers that can be written as palindromic concatenations of two distinct repdigits. Our method combines the theory of linear forms in logarithms of algebraic numbers, together with Baker’s reduction technique.

This sequence has been studied for its rich arithmetic and combinatorial properties, and its connection to balancing numbers, which solve the Diophantine equation: 1 + 2 + · · · + (x − 1) = (x + 1) + · · · + (x + r) for suitable integers x and r. The Balancing numbers also appear in applications to continued fractions and are related to certain classes of Diophantine tuples. The first few terms of the sequence are: 0, 1, 6, 35, 204, 1189, 6930, 40391, . . .

First, we obtain an upper bound for n by comparing the growth of Bn with the number of digits in the target palindrome. Next, we apply Matveev’s theorem on linear forms in logarithms to derive explicit lower bounds. Finally, we employ the Baker–Davenport reduction method to restrict the problem to a finite set of cases, which are then checked computationally. This procedure yields the following result:

Theorem. There is no Balancing number which is a palindromic concatenation of two distinct repdigits.

The primary objective of this article is to extend the fundamental concepts of ring theory to the setting of uniform HX rings and elucidate the structural differences between classical rings and these newly defined algebraic systems. By positioning uniform HX rings within the broader framework of hyperalgebra, this study seeks to advance the generalization of algebraic structures characterized by multivalued operations and contribute to the theoretical expansion of the field. To achieve this aim, this paper introduces and systematically analyzes the notions of subalgebraic structures and the product of uniform HX rings. It investigates the conditions under which these constructions preserve uniformity and examines several algebraic properties that arise from their interaction. Particular emphasis is placed on HX ideals, the images and pre-images of which are characterized through homomorphisms defined between uniform HX rings. This approach not only clarifies the behavior of ideals under structure-preserving mappings but also provides a solid foundation for subsequent theoretical developments. A further objective of this study is to generalize the concept commonly studied as hyperrings in the hyperalgebraic literature. After formally presenting the definition of uniform HX rings, this article establishes their existence through carefully constructed examples, thereby demonstrating the mathematical viability of the proposed structure. Building on this foundation, key properties of uniform HX rings are identified, revealing both their parallels with and departures from classical ring theory. Moreover, HX ring homomorphisms are defined to support the formulation of algebraic structure theorems and illuminate the relationships among uniform HX rings. These mappings play a central role in understanding structural correspondences within the theory. This article culminates in the presentation of a fundamental theorem that ensures a one-to-one correspondence between specific algebraic objects, thereby reinforcing the theoretical framework and offering a basis for future research in hyperalgebra and related areas.

This talk is devoted to the study of a distinguished class of modules characterized by the property that all crumbling submodules of their factor modules are trivial. This defining condition imposes strong internal constraints on the module structure and leads naturally to a coherent and well-organized framework within module theory and relative homological algebra. The motivation behind this investigation arises from the observation that such a restriction allows for a systematic treatment of both structural and homological properties, thereby enabling meaningful comparisons with classical module theory concepts. The primary objective of the talk is to analyze the fundamental structural properties of this class of modules and to examine the relative classes that are naturally induced by it. Within this framework, the proposed module class is positioned among several well-known and extensively studied classes of modules in the literature. The relationships between these classes are explored in a precise and systematic manner, with particular attention given to similarities, differences, and inclusion properties. In particular, the effects of radicals, direct products, and module homomorphisms on the defining properties of the class are examined in detail. This analysis provides insight into the stability of the class under common operations and clarifies the extent to which the defining condition is preserved. Moreover, it sheds light on how these modules interact with classical methods and techniques in module theory. Overall, the results presented in this talk contribute to a deeper understanding of the interplay between structural properties and homological considerations in module theory. By developing and refining the conceptual framework surrounding this module class, the study lays the groundwork for further investigations in relative homological algebra and highlights new directions for research based on the interaction between algebraic constructions and module theory.

Let $K$ be an algebraic number field of degree $n$ with ring of integers $\mathbb{Z}_K$.
The field $K$ is called monogenic if $\mathbb{Z}_K=\mathbb{Z}[\theta]$ for some primitive element $\theta$ of $Z_K$.
Equivalently, $(1,\theta,\ldots,\theta^{n-1})$ forms a $\mathbb{Z}$-basis of $\mathbb{Z}_K$.
Such a basis is called a power integral basis of $\mathbb{Z}_K$.

If the ring $\mathbb{Z}_K$ does not admit any power integral basis, then $K$ is said to be non-monogenic.
The monogeneity of number fields and the construction of power integral bases have been widely studied in both classical and recent research, and many interesting open problems related to these topics remain the subject of active investigation. One of the most important contributions to this problem was made by Gy\H{o}ry, who effectively proved that there exist only finitely many $\mathbb{Z}$-equivalence classes of elements $\theta$, generating a power integral basis of $\mathbb{Z}_K$.

In this talk, for some fixed positive integers $n$, $m$, and $s$, we study the prime common divisors of the indices associated with certain infinite families of number fields defined by quadrinomials of the form $x^n + a x^m + b x^s + c$.
As an application of our results, we provide some explicit conditions on $a$, $b$, and $c$ under which these number fields are not monogenic.
Our approach is based on Ore's classical theorem on the decomposition of primes in number fields, which is formulated using Newton polygon techniques.

In this presentation, a multivector time generator is introduced in which temporal evolution is represented as the action of a multivector on the phase plane, formulated within two-dimensional geometric (Clifford) algebra. Rather than extending time as an additional scalar or complex parameter, the proposed approach reinterprets the time derivative itself as a structured generator whose scalar, vector, and bivector components encode distinct temporal symmetries. Within this framework, bivector components generate symplectic, Hamiltonian evolution and correspond to reversible dynamics, while scalar components produce uniform contraction or expansion in the phase plane, providing a direct geometric characterization of irreversibility. Vector components generate reversible but anti-symplectic transformations, such as reflections, revealing temporal symmetries that lie outside the standard Hamiltonian and complex-time formalisms. This decomposition yields a natural classification of linear dynamical systems into elliptic, hyperbolic, and nilpotent regimes, corresponding to oscillatory, overdamped, and critically damped behavior. General solutions are expressed through multivector exponentials, allowing reversible and irreversible contributions to factorize transparently. The framework further clarifies the status of complex time, showing that it emerges only as a restricted case when vector components vanish and the generator reduces to an even subalgebra admitting a Wirtinger representation. Outside this regime, temporal evolution is intrinsically a multivector and cannot be described by a single complex parameter. By focusing on the algebraic structure of the time generator rather than on time as a primitive parameter, the proposed framework provides a unified geometric language for temporal symmetries in dynamical systems and offers new tools for analyzing reversible and irreversible evolution within a purely algebraic setting.

In 1976, Sato introduced the concept of an almost paracontact structure. On such a manifold, two types of compatible metrics can be considered—the induced transformations, which are isometries, or the antiisometries on the paracontact distribution of the tangent space. The main difference between them lies in the type of associated tensor (0,2) of the metric on the structure. In compatible manifolds, it is a metric, while in metric manifolds, it is a 2-form.

Furthermore, different geometers contributed to the development of the study of the first case manifolds—Riemannian almost paracontact manifolds. In 1980, Sasaki introduced the notion of a Riemannian almost paracontact manifold of type (p,q). By p and q he meant the multiplicities of the eigenvalues ​​+1 and -1 of the structural endomorphism φ.In addition to these eigenvalues, the tensor field φ also has a simple eigenvalue 0.

In the other possible case, i.e., the case of almost paracontact metric manifolds, the metric is pseudo-Riemannian with a signature (n+1, n). These geometric objects have been well studied by a number of authors.

An almost paracontact structure (φ, ξ, η) that possesses a traceless endomorphism φ plays the role of an almost paracomplex structure on the paracontact distribution of a (2n+1)-dimensional smooth manifold and is called a Π-structure. Such a manifold, equipped with a Π-structure, is called a Π-manifold, and when it is also equipped with a Riemannian metric g, it is called a Riemannian Π-manifold.

In this work, a generalization of para-Ricci-like solitons with torse-forming potential on para-Sasaki-like Riemannian Π-manifolds, which is constant multiple of the Reeb vector field, is investigated. Necessary and sufficient conditions are established for such solitons to be equivalent to almost-para-Einstein-like metrics. Further results are derived concerning parallel symmetric covariant tensors of second order. An explicit example in an arbitrary dimension is constructed to illustrate the obtained results.

Topological graph embeddings, particularly 2-cell embeddings, establish a fundamental link between graph theory and surface topology. For a connected graph G, the minimum genus γ(G) and maximum genus γ_M(G) are defined as the smallest and largest genera of orientable surfaces admitting a 2-cell embedding of G, respectively. It is known that an upper bound for the maximum genus of a graph G is
γ_M(G)≤ ⌊β(G)/2⌋, where β(G) is the Betti number of G. A graph is said to be upper embeddable if this equality is achieved.

First, we introduce a lexicographic labeling of the vertices and edges of the Johnson graph J(n,k). Using this labeling, a spanning tree is constructed by sequentially adding edges while avoiding cycles. We prove that this spanning tree is a splitting tree, whose complement contains at most one component with an odd number of edges, and hence establish the upper embeddability of J(n,k) for all n>k≥2 through Jungerman’s characterization. Finally, by applying the classical upper bound in terms of the Betti number, we obtain an explicit formula for the maximum genus of J(n,k).

We prove that the Johnson graph J(n,k) is upper embeddable for all n>k≥2 by explicitly constructing a splitting tree. As a consequence, the maximum genus of J(n,k) is obtained as γ_M(J(n,k))≤ ⌊(k(n-k)-2)/(2k!)∏_{i=1}^{k-1}(n-i)⌋.

In this study, we establish the upper embeddability of the Johnson graph J(n,k) for all n>k≥2. Using an explicit splitting tree together with Jungerman’s characterization, we show that the maximum genus of J(n,k) attains the theoretical upper bound given by its Betti number. This leads to an exact formula for γ_M(J(n,k)).

Cyclic codes form a fundamental class of linear error-correcting codes widely used in digital communications and data storage systems due to their efficient algebraic structure. Despite their long-standing presence in coding theory, the exploration of cyclic codes through the lens of polynomial rings over finite fields continues to reveal new insights into their structural properties and practical applications.

In this study, cyclic codes are represented as ideals of the quotient ring F_q[x]/(xⁿ-1), where F_q denotes a finite field of order q. Ring-theoretic methods are employed to analyze the relationship between codewords and generator polynomials. The factorization of xⁿ-1 is examined to determine the structure of all possible cyclic codes of a given length, and algebraic techniques are used to derive their dimension and minimum distance.

The theoretical analysis demonstrates a direct correspondence between ideals in the quotient ring and cyclic codes, highlighting how generator polynomials uniquely define each code. Several illustrative examples show how this algebraic framework enables systematic construction of cyclic codes with predictable parameters. The results confirm that the polynomial approach simplifies encoding procedures and provides a clear method for calculating essential code characteristics, such as code length, dimension, and error-correcting capability.

This study reinforces the effectiveness of polynomial rings as a unifying algebraic framework for cyclic codes. By connecting abstract ring-theoretic concepts with practical coding applications, the work bridges theory and implementation, offering a foundation for further explorations of advanced cyclic code structures and their applications in modern communications systems. The approach also opens avenues for designing new classes of codes with enhanced error-correcting properties.