Search Results (45)

We characterize all additive biderivations on the incidence algebra $I(P,\mathcal{R})$ of a locally finite poset P over a commutative ring with unity $\mathcal{R}$. By decomposing P into its connected chains, we prove that any additive biderivation splits uniquely into a sum of inner biderivations and extremal ones determined by chain components. In particular, when every maximal chain of P is infinite, all additive biderivations are inner. Full article

In this article, a correspondence between some important classes of numerical semigroups and well-known families of bipartite graphs has been established. Also, it has been demonstrated that, if $\mathrm{m}(\mathrm{S})$ is the multiplicity of a numerical semigroup $\mathrm{S}$, then the diameter and the girth of bipartite gap poset graphs are bounded by the numbers $2\mathrm{m}(\mathrm{S})-3$ and $\mathrm{m}(\mathrm{S})-1$, respectively. Moreover, the Python code to compute the diameter and girth of gap poset graphs has been implemented. Full article

Uninorms are aggregation operators that generalize the t-norms (t-conorms), which are extensions of the logical connectives $\wedge (\vee )$ to the fuzzy set theory. The methods of constructing uninorms on more general algebraic structures (such as bounded posets, lattices, etc.) are an important subject of study, including an extensive work concerning these operations on the unit real interval $[0, 1]$. The construction of uninorms on bounded lattices has been extensively studied using various aggregation functions, such as t-norms, t-conorms, and t-subnorms. In this paper, we present construction methods for uninorms, based on the elements of a lattice, without using the existence of the mentioned operators. We determine the necessary and sufficient conditions for the introduced construction methods to result in the uninorms. Then, we show the differences between our methods and several methods known from the literature, including some illustrative examples. Full article

This study investigates the relationship between sovereign credit rating transitions and domestic equity market performance, focusing on Greece from 2004 to 2024. Although credit ratings are central to sovereign risk assessment, their immediate influence on financial markets remains contested. This research adopts a multi-method analytical framework combining algebraic combinatorics and time-series econometrics. The methodology incorporates the construction of a directed credit rating transition graph, the partially ordered set representation of rating hierarchies, rolling-window correlation analysis, Granger causality testing, event study evaluation, and the formulation of a reward matrix with optimal rating path optimization. Empirical results indicate that credit rating announcements in Greece exert only modest short-term effects on the Athens Stock Exchange General Index, implying that markets often anticipate these changes. In contrast, sequential downgrade trajectories elicit more pronounced and persistent market responses. The reward matrix and path optimization approach reveal structured investor behavior that is sensitive to the cumulative pattern of rating changes. These findings offer a more nuanced interpretation of how sovereign credit risk is processed and priced in transparent and fiscally disciplined environments. By bridging network-based algebraic structures and economic data science, the study contributes a novel methodology for understanding systemic financial signals within sovereign credit systems. Full article

A partially ordered set (or a poset, for short) is a set endowed with a partial order relation, i.e., with a reflexive, anti-symmetric, and transitive binary relation. As mathematical objects, posets have been intensively studied in the last century, coming to play essential roles in pure mathematics, logic, and theoretical computer science. More recently, they have been increasingly employed in data analysis, multi-criteria decision-making, and social sciences, particularly for building synthetic indicators and extracting rankings from multidimensional systems of ordinal data. Posets naturally represent systems and phenomena where some elements can be compared and ordered, while others cannot be and are then incomparable. This makes them a powerful data structure to describe collections of units assessed against multidimensional variable systems, preserving the nuanced and multi-faceted nature of the underlying domains. Moreover, poset theory collects the proper mathematical tools to treat ordinal data, fully respecting their non-numerical nature, and to extract information out of order relations, providing the proper setting for the statistical analysis of multidimensional ordinal data. Currently, their use is expanding both to solve open methodological issues in ordinal data analysis and to address evaluation problems in socio-economic sciences, from multidimensional poverty, well-being, or quality-of-life assessment to the measurement of financial literacy, from the construction of knowledge spaces in mathematical psychology and education theory to the measurement of multidimensional ordinal inequality/polarization. Full article

Let Y be a smooth compact n-manifold. We studied smooth embeddings and immersions $\beta :M\to \mathbb{R}\times Y$ of compact n-manifolds M such that $\beta \left(M\right)$ avoids some priory chosen closed poset $\mathrm{\Theta }$ of tangent patterns to the fibers of the obvious projection $\pi :\mathbb{R}\times Y\to Y$. Then, for a fixed Y, we introduced an equivalence relation between such $\beta$’s; creating a crossover between pseudo-isotopies and bordisms. We called this relation quasitopy. In the presented study of quasitopies, the spaces ${\mathcal{P}}_d^{\mathbf{c}\mathrm{\Theta }}$ of real univariate polynomials of degree d with real divisors, whose combinatorial patterns avoid a given closed poset $\mathrm{\Theta }$, play the classical role of Grassmanians. We computed the quasitopy classes ${\mathcal{Q}}_d^{\mathsf{emb}}(Y,\mathbf{c}\mathrm{\Theta })$ of $\mathrm{\Theta }$-constrained embeddings $\beta$ in terms of homotopy/homology theory of spaces Y and ${\mathcal{P}}_d^{\mathbf{c}\mathrm{\Theta }}$. We proved also that the quasitopies of embeddings stabilize, as $d\to \infty$. Full article

One classic result in domain theory is that the Scott space of every domain (continuous directed complete poset) is sober. Johnstone constructed the first directed complete poset (dcpo for short) whose Scott space is not sober. This non-sober dcpo has been used in many other parts of domain theory and more properties of it have been uncovered. In this survey paper, we first collect and prove the major properties (some of which are new as far as we know) of Johnstone’s dcpo. We then propose a general method of constructing a dcpo from given posets and prove some properties. Some problems are posed for further investigation. This paper can serve as a relatively complete resource on Johnstone’s dcpo. Full article

Following the notions of $\mathrm{\Omega }$-set and $\mathrm{\Omega }$-algebra where $\mathrm{\Omega }$ is a complete lattice, we introduce P-algebras, replacing the lattice $\mathrm{\Omega }$ by a poset P. A P-algebra is a classical algebraic structure in which the usual equality is replaced by a P-valued equivalence relation, i.e., with the symmetric and transitive map from the underlying set into a poset P. In addition, this generalized equality is (as a map) compatible with the fundamental operations of the algebra. The diagonal restriction of this map is a P-valued support of a P-algebra. The particular subsets of this support, its cuts, are classical subalgebras, while the cuts of the P-valued equality are congruences on the corresponding cut subalgebras. We prove that the collection of the corresponding quotients of these cuts is a centralized system in the lattice of weak congruences of the basic algebra. We also describe the canonical representation of P-algebras, independent of the poset P. Full article

By combining a lattice subgroup diagram of ${\mathbb{Z}}_m$ with a weighted poset metric, we introduce a new weighted coordinates poset metric for codes over ${\mathbb{Z}}_m$, called the $LS$-poset metric. When I is an ideal in a poset, the concept of I-perfect codes with an $LS$-poset metric is investigated. We obtain a Singleton bound for codes with the $LS$-poset metric and define MDS codes. When the poset in the poset metric is a chain, we provide sufficient conditions for a code with the $LS$-poset metric to be r-perfect for some $r\in \mathbb{N}$. Full article

In this paper, we express ${\sum }_{n,m\ge 1}{\displaystyle \frac{{\mathit{\epsilon }}_1^n{\mathit{\epsilon }}_2^m{M}_n^{\left(u\right)}{M}_m^{\left(v\right)}}{n^r{m}^s{(n+m)}^t}}$ as a linear combination of alternating multiple zeta values, where ${\mathit{\epsilon }}_i\in \{1,-1\}$ and $M_k^{\left(u\right)}\in \{H_k^{\left(u\right)},{\overline{H}}_k^{\left(u\right)}\}$, with $H_k^{\left(u\right)}$ and ${\overline{H}}_k^{\left(u\right)}$ being harmonic and alternating harmonic numbers, respectively. These sums include Subbarao and Sitaramachandrarao’s alternating analogues of Tornheim’s double series as a special case. Our method is based on employing two different techniques to evaluate the specific integral associated with a 3-poset Hasse diagram. Full article

The concepts of a fuzzy connected set (fc set) and a fuzzy consistently connected set (fcc set) are introduced on fuzzy posets, along with a discussion of their basic properties. Inspired by some equivalent conditions of crisp connected sets, the characterizations of the fc sets are given, and we also explore fuzzy completeness and fuzzy compactness in addition to defining a new fuzzy way-below relation based on fcc complete sets. Using this relationship as a basis, the fcc domain is also provided and studied, and its equivalent characterizations are obtained. In summary, we develop a method to establish fcc completeness from a continuous poset. Full article

In this paper, we solve the open problem posed by Kuba by expressing ${\sum }_{j,k\ge 1}\frac{H_k^{\left(u\right)}{H}_j^{\left(v\right)}{H}_{j+k}^{\left(w\right)}}{j^r{k}^s{(j+k)}^t}$ as a linear combination of multiple zeta values. These sums include Tornheim’s double series as a special case. Our approach is based on employing two distinct methods to evaluate the specific integral proposed by Yamamoto, which is associated with the two-poset Hasse diagram. We also provide a new evaluation formula for the general Mordell–Tornheim series and some similar types of double and triple series. Full article

It is well known that if a poset satisfies Property A and its dual form, then the o-convergence and $o_2$-convergence in the poset are equivalent. In this paper, we supply an example to illustrate that a poset in which the o-convergence and $o_2$-convergence are equivalent may not satisfy Property A or its dual form, and carry out some further investigations on the equivalence of the o-convergence and $o_2$-convergence. By introducing the concept of the local Frink ideals (the dually local Frink ideals) and establishing the correspondence between ID-pairs and nets in a poset, we prove that the o-convergence and $o_2$-convergence of nets in a poset are equivalent if and only if the poset is ID-doubly continuous. This result gives a complete solution to the problem of E.S. Wolk in two modes of order convergence, which states under what conditions for a poset the o-convergence and $o_2$-convergence in the poset are equivalent. Full article

The convergence theorems play an important role in the theory of probability and statistics and in its application. In recent times, we studied three types of convergence of intuitionistic fuzzy observables, i.e., convergence in distribution, convergence in measure and almost everywhere convergence. In connection with this, some limit theorems, such as the central limit theorem, the weak law of large numbers, the Fisher–Tippet–Gnedenko theorem, the strong law of large numbers and its modification, have been proved. In 1997, B. Riečan studied an almost uniform convergence on D-posets, and he showed the connection between almost everywhere convergence in the Kolmogorov probability space and almost uniform convergence in D-posets. In 1999, M. Jurečková followed on from his research, and she proved the Egorov’s theorem for observables in MV-algebra using results from D-posets. Later, in 2017, the authors R. Bartková, B. Riečan and A. Tirpáková studied an almost uniform convergence and the Egorov’s theorem for fuzzy observables in the fuzzy quantum space. As the intuitionistic fuzzy sets introduced by K. T. Atanassov are an extension of the fuzzy sets introduced by L. Zadeh, it is interesting to study an almost uniform convergence on the family of the intuitionistic fuzzy sets. The aim of this contribution is to define an almost uniform convergence for intuitionistic fuzzy observables. We show the connection between the almost everywhere convergence and almost uniform convergence of a sequence of intuitionistic fuzzy observables, and we formulate a version of Egorov’s theorem for the case of intuitionistic fuzzy observables. We use the embedding of the intuitionistic fuzzy space into the suitable MV-algebra introduced by B. Riečan. We formulate the connection between the almost uniform convergence of functions of several intuitionistic fuzzy observables and almost uniform convergence of random variables in the Kolmogorov probability space too. Full article

This paper employs an order-theoretic framework to explore the intricacies of formal concepts. Initially, we establish a natural correspondence among formal contexts, preorders, and the resulting partially ordered sets (posets). Leveraging this foundation, we provide insightful characterizations of atoms and coatoms within finite concept lattices, drawing upon object intents. Expanding from the induced poset originating from a formal context, we extend these characterizations to discern join-irreducible and meet-irreducible elements within finite concept lattices. Contrary to a longstanding misunderstanding, our analysis reveals that not all object and attribute concepts are irreducible. This revelation challenges the conventional belief that rough approximations, grounded in irreducible concepts, offer sufficient coverage. Motivated by this realization, the paper introduces a novel concept: rough conceptual approximations. Unlike the conventional definition of object equivalence classes in Pawlakian approximation spaces, we redefine them by tapping into the extent of an object concept. Demonstrating their equivalence, we establish that rough conceptual approximations align seamlessly with approximation operators in the generalized approximation space associated with the preorder corresponding to a formal context. To illustrate the practical implications of these theoretical findings, we present concrete examples. Furthermore, we delve into the significance and potential applications of our proposed rough conceptual approximations, shedding light on their utility in real-world scenarios. Full article