Let [Formula: see text] and [Formula: see text] be the biHecke monoids associated with Coxeter groups of types [Formula: see text] and [Formula: see text], respectively. In this paper, we explore the finite basis problems for biHecke monoids [Formula: see text] and [Formula: see text] and their Borel submonoids [Formula: see text] and [Formula: see text]. It is shown that [Formula: see text] and [Formula: see text] are inherently non-finitely based for all [Formula: see text], [Formula: see text] is hereditarily finitely based and [Formula: see text] is non-finitely based.
Let [Formula: see text] be a simple graph on [Formula: see text] vertices and [Formula: see text] denote the corresponding binomial edge ideal in [Formula: see text], where [Formula: see text] is a field. In this paper, we provide an upper bound for the regularity of binomial edge ideals of trees. Our approach leverages the recently developed framework of Betti splittings of binomial edge ideals, a powerful technique for studying homological invariants via decompositions into simpler ideals. We also give a lower bound for the same. As a consequence, we explicitly compute the regularity of binomial edge ideals of trees under certain conditions.
This paper is devoted to the generalizations of weak Drazin inverses from square matrices to the broader framework of generalized Drazin invertible operators. Within this setting, we establish several equivalent characterizations of the newly defined inverses. Furthermore, we introduce the m-minimal rank generalized weak Drazin inverse as a novel extension of the classical definition. By employing suitable block-matrix decompositions, we derive explicit matrix representations for these operators. These representations prove instrumental in deriving new theoretical results and establish a unified framework for the study of weak Drazin inverses in operator theory.
Let [Formula: see text] be a unital ∗-algebra containing the nontrivial idempotents, and let [Formula: see text] be a family of nonlinear maps satisfying [Formula: see text] where [Formula: see text] and [Formula: see text] for all [Formula: see text] and [Formula: see text]. In this article, we prove that [Formula: see text] is an additive higher ∗-derivation. As applications, nonlinear mixed bi-skew Jordan-Lie-type higher derivations on prime ∗-algebras, factor von Neumann algebras and standard operator algebras are characterized.
This paper is devoted to the study of a natural generalization of the positive Witt algebra. We first show that the algebra under consideration can be realized as a mutation of the positive Witt algebra. We then construct its pro-solvable Lie extensions and analyze their structural properties. Moreover, we prove that these extensions constitute a family of complete pro-solvable Lie algebras.
We study the zero-divisor graph [Formula: see text] attached to a commutative ring R and an unfaithful R-module M, with vertices the nonzero zero-divisors of R modulo I = [Formula: see text] and adjacency defined by annihilation on M. Starting from [Moh’d and Ahmed, Extending the Anderson–Livingston zero-divisor graph via unfaithful modules, Appl. Analysis Discrete Math, (2026)], we develop an exact fiber-decomposition theory that realizes [Formula: see text] as a mixed blow-up of [Formula: see text]. This viewpoint yields explicit formulas for the triangle number, clique number, independence number, chromatic number, diameter, girth, domination number, and several parity properties whenever I is finite. In particular, we characterize bipartiteness, regularity, and Eulerian behavior, and derive computable invariants from the quotient graph. Several concrete examples over [Formula: see text] and truncated polynomial rings illustrate the theory, while comparison tables and figures show how the annihilator ideal controls the passage from [Formula: see text] to [Formula: see text]. Our results suggest new problems on domination and planarity.
Let D be a k-edge weighted VEW graph and I = I (D) be its edge ideal. In this paper, we characterize the irredundant irreducible decomposition of edge ideal I. Also, we study the Cohen-Macaulay property and Castelnuovo-Mumford regularity of different classes of k-edge weighted VEW graphs.
Let [Formula: see text] be an abelian category. Denote by [Formula: see text] the bounded derived category of [Formula: see text]. In this paper, we investigate the lower bounds for the levels of objects in [Formula: see text] with respect to a (co)resolving subcategory satisfying a certain condition. As an application, we not only recover the results of Altmann–Grifo–Montaño–Sanders–Vu, and Awadalla–Marley but also extend them to establish lower bounds for levels with respect to some other subcategories in an abelian category.
In this short note we show a sufficient condition for two subgroups of a finite group being conjugate from their local conjugacy.
In this paper, we first introduce the notions of extended Leibniz-dendriform algebras and extended post-Leibniz algebras, and investigate their intrinsic relationship with extended Rota-Baxter Leibniz algebras. Next, we propose the concepts of extended Rota-Baxter Leibniz bialgebras and admissible quadruples of extended Rota-Baxter Leibniz algebras. Furthermore, we define the Manin triple and the matched pair of extended Rota-Baxter Leibniz algebras. We then prove that extended Rota-Baxter Leibniz bialgebras can be equivalently characterized by matched pairs and Manin triples of extended Rota-Baxter Leibniz algebras. Finally, we construct several classes of extended Rota-Baxter Leibniz bialgebras via the admissible classical Leibniz Yang-Baxter equation (cLYBe) and 𝒪-operators.
In this paper, we prove that the intersection of all non-nilpotent maximal subgroups of a non-solvable group containing the normalizer of some Sylow subgroup is nilpotent, which provides an extension of Shidov’s theorem and Shlyk’s theorem.
In this paper, we study a class of irreducible uniformly bounded weight modules for map extended Special Lie algebra [Formula: see text], where the elements [Formula: see text] and [Formula: see text] act nontrivially on such modules for some nonzero [Formula: see text]. It is shown that these modules are irreducible for the underlying extended special algebra [Formula: see text] with finite-dimensional weight spaces, which are precisely the irreducible Jet modules (with some rescaled variable) for the Lie algebra [Formula: see text] with finite-dimensional weight spaces.
This article investigates the geometric structure of the q-Berezin range of operators on the Hardy space, with a particular emphasis on convexity. Convexity results are established for several operator classes, including Toeplitz, weighted shift, and certain composition operators. The study demonstrates the relationship between the q-Berezin range and the classical Berezin range.
Let & gfr; be the Lie superalgebra of polynomial vector fields on & Copf;(1|1) and & ell; is an element of & Zopf;(>= 1). In this paper, we first study the Kac module over a finite-dimensional Lie superalgebra & gfr;(>= 0)/& gfr;(>=& ell;), which is an induced module from a simple module over a certain finite-dimensional Lie algebra. We determine the simplicity of such module, and its submodules and quotient modules when it is not simple. Then based on these results, we classify all simple smooth & gfr;-modules. We also give a characterization of simple smooth & gfr;-modules. In addition, we describe all Whittaker vectors and submodules of the universal Whittaker & gfr;-module.
This paper introduces the concept of classical S-primary submodules, a generalization of S-primary submodules where S subset of R is a given multiplicatively closed subset. It provides various examples, characterizations, and relationships with classical primary submodules. Additionally, it investigates the conditions under which a submodule qualifies as classical S-primary. The paper proves the existence of classical S-primary decomposition in S-Noetherian modules, that is, every proper submodule E of an S-Noetherian R-module X with (E :(R) X) boolean AND S = empty set can be expressed as a finite intersection of classical S-primary submodules. Finally, the behavior of the concept has been examined under homomorphisms, in factor modules, Cartesian products of modules, and amalgamated duplication of modules.
Let M be a finite von Neumann algebra with no central summands of type I-1. If delta : M -> M is a nonlinear 2-local Lie n-centralizer with n >= 2, then delta(a) = za + tau(a) for all a is an element of M, where z is an element of & Zscr;(M) and tau : M ->& Zscr;M is a mapping which annihilates each (n - 1)th commutator of M.
We introduce and study sum-difference prime (sd-prime) ideals in noncommutative rings, extending the square-difference absorption property from the commutative setting. We establish fundamental relationships between sd-prime, prime, and semiprime ideals, and we analyze the stability of sd-primeness under standard ring constructions, obtaining structural classification theorems for sd-prime ideals in a wide range of related rings, including homomorphic images, quotients, matrix rings, direct products, trivial ring extensions, upper triangular matrix rings, and pullback rings. We further introduce noncommutative amalgamations, where we describe the ideal structure and obtain preservation results for sd-prime ideals, supported by explicit counterexamples. In the commutative setting, we develop an ideal-theoretic geometry for sd-prime ideals by introducing the sd-spectrum SdSpec(R) together with its sd-Zariski topology and the associated sd-radical, establishing spectral properties and identifying conditions under which closed sets admit unique generic points.
In this paper, we introduce the generalized right e-core inverse in Banach algebras. By employing a decomposition involving quasinilpotent elements, we characterize this newly generalized inverse, establish its relationship with the right g-Drazin inverse, and derive its representations. As an application, we obtain new characterizations of the m-core-EP inverse in Minkowski spaces.
In this paper, we find that the cyclotomic KLR algebra of type A and blocks of the cyclotomic (degenerate) affine Hecke algebra of a symmetric group can be realized as the same quotient of an algebra. This can be seen as a re-proof of the Brundan-Kleshchev-Rouquier isomorphism. Our approach is based on the computations in suitable localization of the affine Hecke algebras by following Rouquier, and our technique is based on Brundan and Kleshchev's original proof.
