Consider the graph on the monoid (under v-multiplication) Fv(R) of divisorial fractional ideals of an integral domain R such that distinct I and J are adjacent if and only if there exist divisorial A and B such that A is covered (under inclusion) in Fv(R) by I and J, and B covers both I and J. The connected component [R] containing R is investigated, and is shown to be a subgroup of Fv(R) for several classes of integral domains, including Krull domains, Prüfer domains, GCD domains, and pseudovaluation domains. In fact, among the main results, it is proved that R is a Krull domain if and only if [R] is a group such that the quotient monoid Fv(R)/[R] is isomorphic to the group of integers. Further characterizations are given for the special cases of Dedekind domains and unique factorization domains.
A semigroup S is called right coherent if every finitely-generated subact of every finitely-presented right S-act is finitely-presented. Left coherency is defined dually, and S is coherent if it is both right and left coherent. In this article we show that graph inverse semigroups are coherent. Such semigroups were introduced by Ash and Hall and are closely connected to C⁎-algebras, Cohn path algebras and Leavitt path algebras.
In this paper, we study prime deductive systems in unital lattice-ordered quantum B-algebras. Assuming the lattice of deductive systems is distributive, we first prove prime extension and intersection representation theorems. We then endow the set P(X) of all proper prime deductive systems with a Zariski topology and investigate its fundamental properties: it is a T0 -space, and under the same distributivity assumption, the subspace M(X) of maximal deductive systems is a T1 -space. In the finite case, we show this space P(X) is spectral. Furthermore, this topology yields a canonical Galois adjunction between the lattice of deductive systems and the lattice of open subsets of P(X), leading to a closure operator that characterizes the spectrally closed deductive systems as its fixed points.
In this paper we develop an approach to the full quaternionic moment problem. We define a hierarchy Hk[q⁎,q]⊂Hk+1[q⁎,q], k∈N0∪{∞}, of two-sided H-linear spaces of quaternionic polynomials which are invariant under conjugation of quaternions and determine the hermitian parts of these spaces explicitly. Using a generalization of Choquet's theorem on adapted spaces to quaternions we provide necessary and sufficient solvability criteria for the quaternionic moment problem of each space Hk[q⁎,q]. The hermitian part of H∞[q⁎,q] is the real polynomial algebra R[x0,x1,x2,x3]. This enables us to apply real algebraic geometry (Positivstellensätze) to the quaternionic moment problem on H∞[q⁎,q].
Many open conjectures in the representation theory of finite groups can be studied by reducing them to related questions about quasi-simple groups. In such studies, p-radical subgroups typically play a critical role. To classify the p-radical subgroups of a finite group, we can first classify the elementary abelian p-subgroups and find their local structure. In this paper, we use the classification and local structure of the nontoral elementary abelian subgroups of PGLn(C) to find the splitting of the conjugacy classes of these subgroups, as well as their local structure in PGLn(q) and PGUn(q). The main cases we treat are distinguished by whether the centralizers of these subgroups are connected or disconnected, and by whether the Steinberg endomorphism acts trivially or nontrivially on the corresponding normalizer quotient.
We introduce the twisted super Yangian Yıs of quasi-split type A in the Drinfeld current presentation for an arbitrary symmetric parity sequence s. We prove via Gauss decomposition that the twisted super Yangian Yıs is isomorphic to the (special) twisted super Yangian Ys, previously defined via the R-matrix presentation. As a corollary, we prove that the twisted super Yangians Yıs corresponding to different parity sequences with the same m and n are isomorphic. Additionally, we establish the PBW theorem for Yıs and describe the center of Ys in terms of Gaussian generators, thereby generalizing known results for the nonsuper quasi-split type A case.
We give a geometric model for the non-τ-rigid modules over acyclic path algebras of type D˜n. Similar models have been provided for module categories over path algebras of types An,Dn, and A˜n as well as the τ-rigid modules of type D˜n. A major draw of these geometric models is the “intersection-dimension formulas” they often come with. These formulas give an equality between the intersection number of the curves representing the modules in the geometric model and the dimension of the extension spaces between the two modules. This formula allows us to calculate the homological data between two modules combinatorially. Since there are infinitely many distinct homogeneous stable tubes in the regular component of the Auslander-Reiten quiver of type D˜n, all of which are disjoint, our geometric data requires an extra decoration on the admissible edges in our geometric model to prevent intersections between curves corresponding to modules in distinct stable tubes of the Auslander-Reiten quiver.
A subgroup T of a group Xis almost maximal if NX(T) is a maximal proper subgroup of X. We explore this notion when T is a finite simple group and Xis a finite alternating group or a finite symmetric group. We prove that if T is almost maximal and NX (T) is not primitive, then T must be a cyclic group of order at most three or a finite alternating group. On the other hand we show that, for every finite nonabelian simple group T with the single exception of PSL(2, 11), there exists n such that T is an almost maximal subgroup of an alternating group An with T acting primitively, and moreover there is such a primitive embedding with NAn (T) the unique maximal subgroup of An containing T. Several open problems are posed. (c) 2026 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons.org/licenses/by/4.0/).
Given two subsets X,Y of a finite group G, we write Pr(X,Y) for the probability that random elements x∈X and y∈Y commute. If X,Y are subgroups, we denote by Pr⁎(X,Y) the maximum real number ϵ with the property that for every pair of distinct primes p∈π(X) and q∈π(Y) there is a Sylow p-subgroup P of X and a Sylow q-subgroup Q of Y such that Pr(P,Q)≥ϵ.In this paper we handle, among other things, finite groups G with high probabilities Pr⁎(T,G), where T is either a term of the lower central series of G or the generalized Fitting subgroup Fi⁎(G). Our main results show that the structure of such groups is similar, in some precise sense, to that of nilpotent groups.
By work of Howlett and Muraleedaran–Taylor, a parabolic subgroup of a real or complex reflection group always admits a complement in its normalizer. In this note, we investigate this phenomenon for quaternionic reflection groups. Here, in contrast to the real and complex setting, we find that complements of parabolic subgroups do not exist in general. Indeed, there are infinitely many examples of quaternionic reflection groups in arbitrary rank greater than 2 with a parabolic subgroup that does not admit a complement in its normalizer. We give a full classification of parabolic subgroups of irreducible quaternionic reflection groups and describe their complements, if the latter exist.
We give a geometric realization of the whole universal enveloping algebras of Borcherds-Bozec algebras using quivers with loops via the motivic semi-derived Hall algebra approach.
Let R be a standard graded polynomial ring over a field k. The paper focuses on homogeneous ideals J⊂R of codimension 2 generated by three forms of the same degree d≥2 that are almost Cohen–Macaulay, i.e., of homological dimension 2. Based on the structure of the minimal graded free resolution of J and numerical data inspired by the corresponding shifts, one introduces the notion of a level matrix associated with these data. The main result provides a complete characterization of an almost Cohen–Macaulay 3-generated ideal J of codimension 2 in terms of the existence of a related level matrix for which J arises as the ideal of its maximal minors that fix a submatrix. One provides algebraic and geometric examples illustrating the results.
The proof of the inductive McKay condition has been shown to imply that the character theory above the characters of degree not divisible by p of a normal subgroup is locally determined. In this note, we establish a similar result for the Isaacs' head characters of a normal solvable subgroup of an arbitrary group. In particular, we give a new lower bound of the number of conjugacy classes of a finite group in terms of the Carter subgroups of any of its normal solvable subgroups.
Matrix mutation of skew-symmetrizable matrices is foundational in cluster algebra theory. Effective mutation invariants are essential for determining whether two matrices lie in the same mutation class. Casals introduced a binary mutation invariant for skew-symmetric matrices. In this paper, we extend Casals' construction to the skew-symmetrizable setting. When the skew-symmetrizer d_1,…, d_n is pairwise coprime, we obtain two distinct extensions of this invariant.
Let A be an n×n real Toeplitz matrix satisfying A+A⊤=2Jn, where Jn is the all-ones matrix. If Ar(i,j) denotes the r×r contiguous submatrix of A consisting of rows i,i+1,…,i+r−1 and columns j,j+1,…,j+r−1, then for every n≥2 one hasdetAn−1(1,2)+detAn−1(2,1)=2detAn−1(1,1). This confirms a conjecture of Charles R. Johnson (2003). The proof combines a rank-one determinant expansion with Dodgson's condensation formula, and then invokes a polynomial-identity argument in the Toeplitz parameters: after obtaining an equality of squares in the integral domain Z[b1,…,bn−1], we factor it to deduce an identity up to sign and determine the sign by a suitable specialization. We also give an extension of the Bayat–Teimoori arithmetic–geometric mean identity: for every real accretive matrix A, one has the sharp inequalitydetAn−1(1,1)detAn−1(2,2)≥|detAn−1(1,2)+detAn−1(2,1)2|, with equality whenever the symmetric part has at most rank one, i.e. A+A⊤=αww⊤ for some α∈R and w∈Rn∖{0}, recovering the Bayat–Teimoori equality as a special case.
In this paper, we mainly discuss how to use dendriform D-bialgebras to construct Lie bialgebras and the relationship between the solutions of their corresponding Yang-Baxter equations. We provide two methods for obtaining Lie algebras from dendriform algebras using the tensor product with perm algebras, one by means of associative algebras and the other by means of pre-Lie algebras. We elevate both approaches to the level of bialgebras and prove that the Lie bialgebras obtained using these two approaches are the same. There is a correspondence between symmetric solutions of the dendriform Yang-Baxter equation in dendriform algebras and certain skew-symmetric solutions of the classical Yang-Baxter equation in the Lie algebras induced from the dendriform algebras. The connections between triangular bialgebra structures and O-operators related to the solutions of these Yang-Baxter equations are discussed in detail. During the discussion, we also present a method for constructing an infinite-dimensional antisymmetric infinitesimal bialgebra by using the affinization of dendriform D-bialgebras.
We extend Grood's tableau construction of irreducible representations of the rook monoid and Steinberg's analogous result for the full transformation monoid. Our approach is characteristic-free and applies to any submonoid M(n) of the partial transformation monoid on an n-element set that contains the symmetric group. To achieve this, we introduce and study a functor from the category of rational representations of the monoid of n×n matrices to the category of finite dimensional representations of M(n). We establish two branching rules. Our main results describe graded module structures of orbit harmonics quotients for the rook, partial transformation, and full transformation monoids. This yields analogs of the Cauchy decomposition for polynomial rings in n×n variables.
For a p-permutation equivalence between two block algebras of finite groups, we introduce new square diagrams that link the p-permutation equivalence via the Brauer construction to local equivalences between stabilizers of corresponding Brauer pairs. These diagrams can be viewed as lifts of the square diagrams in the definition of isotypies. The proof of the commutativity requires new technical tools, namely a formula for how taking fixed points commutes with extended tensor products of finite sets with group actions and how the Brauer construction commutes with taking extended tensor products of p-permutation modules. These fundamental formulas, generalizing earlier results by Boltje-Danz and by Boltje-Perepelitsky, should be of independent interest.
