In [5], it has been proved that any ring satisfying the identity \((x y) z=y(z x)\) is 5-nice. In this paper, we show that this index of niceness can be reduced when we suppose that the ring also satisfies another identity such that \((x y) z=(x z) y\). In a wider sense, we assume *Corresponding author > that \(A\) is an algebra which is not associative, defined over a field \(K\), which satisfies the identities \((x y) z=y(z x)\) and \((x y) z=\alpha(x z) y\) with \(\alpha \in K\), and we prove that \(A\) is 3-nice if \(\alpha \notin\{-1,1\}\) and 4-nice otherwise.
Let $G$ be a finite group and $\lambda_n(G)$ denote the probability that $n$ randomly chosen elements generate $G$. Let $E(G)$ denote the expected number of random selections required to obtain a generating set of $G$. In this paper, we investigate these functions for certain finite nilpotent groups, with particular attention to non-abelian $p$-groups of order $p^4$. Using subgroup structures, Eulerian functions, and probabilistic methods, we derive explicit formulas for $\lambda_n(G)$ and $E(G)$ for several classes of nilpotent groups. In particular, we obtain generation probabilities and expectation formulas for the non-abelian group $Z_{p^2} \rtimes Z_{p^2}$ and related examples. These results extend previous work on probabilistic generation of finite groups and provide further insight into the relationship between subgroup structure and generation behavior.
We investigate the flat-foldability of 2 × $n$ map folding patterns through the lens of order theory and linear algebra. We model the problem as determining the existence of a linear extension of a poset compatible with topological constraints, viewing the 2 × $n$ map folding as an order extension of the 1 × 2$n$ strip folding. Building on the fact that the configuration space of a strip forms a lattice, we treat the valid map foldings as a specific subset defined by geometric obstructions. To rigorously detect these obstructions (self-intersections), we introduce an algebraic invariant over the field $\mathbb{F}_2$. We represent the overlapping order as a transitive closure matrix over the Boolean semiring and formulate the intersection-avoidance condition as the vanishing of a bilinear form over $\mathbb{F}_2$. This framework yields a purely combinatorial and linear-algebraic characterization of flat-foldability, allowing for the efficient enumeration of all valid 2 × $n$ patterns by filtering the strip lattice.
In this paper, we establish relations defining the coefficients of Roselle polynomials and give their combinatorial interpretation. Introducing the non-ascending permutations, we determine the -Roselle polynomials as well as their -exponential generating functions.
In this work, we investigate the exponential Diophantine equation \[ a^x + (ab+3)^y = b^z, \] where $a,b > 1$ are fixed integers and $x,y,z$ are positive integers. Our approach uses modular arithmetic, divisibility arguments, and the structure of multiplicative groups modulo integers. In the coprime case $\gcd(a,b) = 1$, we show that the equation leads to the congruence \[ 3^y \equiv b^z \pmod{a}, \] which can be interpreted in the finite abelian group $(\mathbb{Z} / a \mathbb{Z})^{\times}$. This perspective reveals that the cyclic subgroups generated by the residue classes of 3 and $b$ must share a common element whenever a solution exists. We also consider the case $a \mid b$, where divisibility and 3-adic valuation arguments impose strong restrictions on the form of $a$ and on the possible values of the exponents. The results obtained provide necessary conditions for the existence of solutions and illustrate how group-theoretic ideas and valuation methods can be effectively applied in the study of exponential Diophantine equations.
Let $(N_n)_{n \ge0}$ be the Narayana’s cows sequence and $(P_n)_{n \ge0}$ the Padovan sequence. The aim in this paper is to study and completely solve the Diophantine equation $N_m=P_n$, where $m$ and $n$ are positive integers.
Let $(N_n)_{n \ge0}$ be the Narayana’s cows sequence and $(T_n)_{n \ge0}$ the Tribonacci sequence. In this paper, we investigate Diophantine equations involving the Tribonacci sequence and the Narayana sequence. More precisely, we solve the Diophantine equations \[T_k=N_mN_n,\] \[N_k=T_mT_n.\] Using a combination of linear forms in logarithms, technical reduction and computational arguments we prove that these equations admit only finitely many solutions. All solutions are explicitly determined.
This article concerns commutative and non-associative algebras, over an infinite field of characteristic $\neq 2$, satisfying the $\omega$-polynomial identity $x^2 x^3=\omega(x) x^4$. We determine the conditions under which such algebras are power-associative or Jordan algebras. Furthermore, we show that they are Bernstein algebras of order at most 3 and also principal train algebras but not necessarily special train algebras. This paper advances the understanding of the algebraic structure of weighted genetic algebras.
In this paper, we introduce a new class of flattened partitions under specific combinatorial constraints, which we call triangular flattened partitions. We describe the structure of this class of flattened partitions containing exactly the pattern $12\cdots r$ and provide recurrence relations that define them according to the kind, the number of runs, and the length of the last run. Finally, we derive the corresponding generating functions. We investigate two-run triangular flattened partitions more thoroughly.
In this work, we develop an asymptotic theory for axe-filtrations on modules, a structure that generalizes classical filtrations, quasi-graduations, and axe quasi-graduations. Our objective is to extend the theory of analytic spread, originally introduced by Northcott and Rees for ideals, to this broader framework. Our approach follows the successive generalizations investigated by Diagana, Brou, and Kablam [1-5, 8]. Our first contribution is the classification of these structures into I-adic, I-good, and f-good types, building upon the fundamental results on quasi-graduations [9, 10]. The main result of this paper establishes a Hilbert-Samuel type theorem for these new structures: we prove that the Hilbert function associated with a good axe-filtration on a finitely generated module is of polynomial type for sufficiently large integers. This fundamental property allows us to rigorously define the analytic spread of an axe-filtration, denoted by lambda(phi), and to establish its primary properties, including its relationship with the module height and the classical analytic spread. This contribution offers new perspectives for asymptotic analysis in algebraic structures governed by constraints where standard filtration theory is no longer applicable.
We complete the classification of isotropy subgroups for the Valentiner action of $A_6$ on $\mathbb{P}^2$. The non-cyclic isotropy subgroups are exactly the dihedral groups $D_6, D_8, D_{10}$ where, throughout this paper, $D_m$ denotes the dihedral group of order $m$. These groups fix isolated points of the invariant sextic curve. The Klein four group is a subgroup of a $D_8$ point stabiliser, but is not the full stabiliser of any point, while $E_9$ and $A_5$ do not occur as point stabilisers. As an application, we describe the singularities of the quotient $\mathbb{P}^2 / A 6 \cong \mathbb{P}(1,2,5)$.
In this paper, we study the structure of a class of commutative, not necessarily associative, algebras that satisfy the o-polynomial identity 2 x2x5 = o x 2x5 + o x 5x2 Using the technique of linearization, ( ) ( ) . we show that if an algebra satisfies such an identity, and has a non-zero idempotent, then it admits a Peirce decomposition. We also establish the links between this class of algebras and other classes of algebras such as Bernstein algebras, power-associative algebras, Jordan and principal train algebras of rank 3.
Let $V$ be a braided vector space of diagonal type. Let $\mathfrak{B}(V), \quad \mathfrak{L}(V)_R$ and $\mathfrak{L}(V)_L$ be Nichols Lie algebra, Nichols $L$-bicharacter algebra and Nichols $R$-bicharacter algebra of $V$, respectively. We show that a monomial belongs to $\mathfrak{L}(V)_R$ (or $\mathfrak{L}(V)_L$ ) if and only if this monomial is $R$-connected (or $L$-connected). We obtain explicit bases for $\mathfrak{L}(V)_R$ and $\mathfrak{L}(V)_L$ over the quantum linear space $V$ with $\operatorname{dim} V=n$.
We classify all non-maximal ordersO f = Z + fO K of ring class number two in the Heegner fields. Exactly ten pairs (K, f)occur. We then treat the smallest example, Z[ -8], and prove that 9=3 & sdot;3=(1 + -8)(1--8) with all four factors being irreducible. These orders provide explicit test cases for future catenary-degree calculations.
We establish parallel-sum analogues of Ky Fan’s determinant inequality when the usual matrix addition is replaced by the parallel sum of positive definite matrices. We also investigate several variants of this parallel Ky Fan inequality, including a scalar determinant estimate, and an analogue of the determinant difference inequality.
We study the localization of strongly hopfian and strongly cohopfing objects in the category of complex sequences of left $A$-modules $COMP(A-Mod)$. Findings include: (1) Let $C$ be an object of $COMP(A-Mod), E$ be a subcomplex of $C$ and $S$ be a saturated multiplicative part of $A$ satisfying the left Ore conditions. If $S_C^{-1} C$ is strongly hopfian and $S_C^{-1} E$ is a direct summand, then $\left(S_C^{-1} C\right) /\left(S_C^{-1} E\right)$ and $S_C^{-1} E$ are both strongly hopfian objects of $COMP\left(S^{-1} A-Mod\right)$. (2) Let $C$ be an object of $COMP(A-Mod), E$ be a subcomplex of $C$ and $S$ be a saturated multiplicative part of $A$ satisfying the left Ore conditions. If $S_C^{-1} E$ is fully invariant, $\left(S_C^{-1} C\right) /\left(S_C^{-1} E\right)$ and $S_C^{-1} E$ are strongly hopfian objects of $COMP\left(S^{-1} A-Mod\right)$, then so is $S_C^{-1} C$. (3) Let $C$ be an object of $COMP(A-Mod), E$ be a subcomplex of $C$ and $S$ be a saturated multiplicative part of $A$ satisfying the left Ore conditions. If $S_C^{-1} C$ is strongly cohopfian and $S_C^{-1} E$ is a direct summand, then $\left(S_C^{-1} C\right) /\left(S_C^{-1} E\right)$ and $S_C^{-1} E$ are both strongly cohopfian objects of $COMP\left(S^{-1} A-Mod\right)$. (4) Let $C$ be an object of $COMP(A-Mod), E$ be a subcomplex of $C$ and $S$ be a saturated multiplicative part of $A$ satisfying the left Ore conditions. If $S_C^{-1} E$ is fully invariant, $\left(S_C^{-1} C\right) /\left(S_C^{-1} E\right)$ and $S_C^{-1} E$ are strongly cohopfian objects of $COMP\left(S^{-1} A-Mod\right)$, then so is $S_C^{-1} C$. (5) Let $C$ be an object of $COMP(A-Mod)$ and $S$ be a saturated multiplicative part of $A$ satisfying the left Ore conditions. If $S_C^{-1} C$ is quasi-injective and strongly hopfian then $S_C^{-1} C$ is a strongly cohopfian object of $COMP\left(S^{-1} A-M o d\right)$. (6) Let $C$ be a complex sequence and $S$ be a saturated multiplicative part of $A$ satis fying the left Ore conditions. If $S_C^{-1} C$ is quasi-projective and strongly cohopfian, then $S_C^{-1} C$ is a strongly hopfian object of $COMP\left(S^{-1} A-Mod\right)$. (7) Let $C$ be an object of $COMP(A-Mod)$ and $S$ be a saturated multiplicative part of $A$ satisfying the left Ore conditions. If $S_C^{-1} C$ is quasi-injective and strongly cohopfian or quasiprojective and strongly hopfian, then $S_C^{-1} C$ is a Fitting complex sequence of left $S^{-1} A$ modules. Received: November 28, 2025Revised: March 19, 2026Accepted: March 30, 2026
In this paper, the notion of hybrid structure is applied to the ideal in Bd-algebras. In fact, we introduce the notions of hybrid Bd-ideal, $P$-ideal, hybrid $P$-ideal and investigate their related properties. In particular, it provides a precise characterization of hybrid Bd-subalgebras and hybrid Bd-ideals, including necessary and sufficient conditions for a hybrid structure to form a Bd-ideal. Moreover, a formal link between hybrid Bd-ideals and $p$-ideals in Bd-algebras has been identified, providing a natural extension of the current results in the theory. These findings contribute to a deeper understanding of the structural properties of Bd-algebras and open new directions for further research in algebraic and logical system.
Numerical semigroups form a fundamental class of additive submonoids of non-negative integers that are closed under addition and have a finite complement in $$\mathbb{N}.$$ Due to their rich algebraic structure and profound combinatorial properties, numerical semigroups play a significant role in many mathematical disciplines, including Commutative Algebra, Coding Theory, Algebraic Geometry, and Singularity Theory. The investigation of the geometric structure of algebraic curves through their associated numerical semigroups occupies central place in this study which is closely connected with the concept of the numerical semigroup type. This study provides a comprehensive analysis of numerical semigroups with determination number five, aimed at to offer a complete characterization of their type sequences.
We prove that a complex-valued function on a finite group defined as the number of ways a group element of a finite group can be written as a commutator representing a character. Also, we show that this function can be written as the sum of irreducible character products with algebraic integers. The motivation is to give more details and extend a result of Frobenius in the setting of a modern character theory.
This work gives the set of algebraic points of a given degree over $\mathbb{Q}$ on the affine curve $C_{a, b}: y^2=a\left(x^5-b^5\right)$ with $(a ; b) \in\{(3 ; 1),(1 ; 3)\}$. It generalizes results by Mulholland [1] and Siksek [2], who described the set of $\mathbb{Q}$-rational points on this curve.
