CyberSec Research

Let $R$ be a finite ring with identity. The clean graph $Cl(R)$ of a ring $R$ is a graph whose vertices are pairs $(e, u)$, where $e$ is an idempotent element and $u$ is a unit of $R$. Two distinct vertices $(e, u)$ and $(f, v)$ are adjacent if and only if $ef = fe = 0$ or $uv = vu = 1$. The graph $Cl_2(R)$ is the induced subgraph of $Cl(R)$ induced by the set $\{(e, u): e \text{ is a nonzero idempotent and } u \text{ is a unit of } R\}$. In this study, we present properties that arise from the isomorphism of two clean graphs and conditions under which two clean graphs over direct product rings are isomorphic. We also examine the structure of the clean graph over the ring $M_2(\mathbb{Z}_p)$ through their $Cl_2$ graph.

Given k similarity classes of invertible matrices, the Deligne-Simpson problem asks to determine whether or not one can find matrices in these classes whose product is the identity and with no common invariant subspace. The author conjectured an answer in terms of an associated root system, and proved one implication in joint work with Shaw. In this paper we prove the other implication, thus confirming the conjecture.

In this paper, we investigate the structural and characterizing properties of the so-called {\it 2-UQ rings}, that are rings such that the square of every unit is the sum of an idempotent and a quasi-nilpotent element that commute with each other. We establish some fundamental connections between 2-UQ rings and relevant widely classes of rings including 2-UJ, 2-UU and tripotent rings. Our novel results include: (1) complete characterizations of 2-UQ group rings, showing that they force underlying groups to be either 2-groups or 3-groups when $3 \in J(R)$; (2) Morita context extensions preserving the 2-UQ property when trace ideals are nilpotent; and (3) the discovery that potent 2-UQ rings are precisely the semi-tripotent rings. Furthermore, we determine how the 2-UQ property interacts with the regularity, cleanness and potent conditions. Likewise, certain examples and counter-examples illuminate the boundaries between 2-UQ rings and their special relatives. These achievements of ours somewhat substantially expand those obtained by Cui-Yin in Commun. Algebra (2020) and by Danchev {\it et al.} in J. Algebra \& Appl. (2025).

It is natural to consider extending the typical construction of relative Poisson algebras from commutative differential algebras to the context of bialgebras. The known bialgebra structures for relative Poisson algebras, namely relative Poisson bialgebras, are equivalent to Manin triples of relative Poisson algebras with respect to the symmetric bilinear forms which are invariant on both the commutative associative and Lie algebras. However, they are not consistent with commutative and cocommutative differential antisymmetric infinitesimal (ASI) bialgebras as the bialgebra structures for commutative differential algebras. Alternatively, with the invariance replaced by the commutative $2$-cocycles on the Lie algebras, the corresponding Manin triples of relative Poisson algebras are proposed, which are shown to be equivalent to certain bialgebra structures, namely relative PCA bialgebras. They serve as another approach to the bialgebra theory for relative Poisson algebras, which can be naturally constructed from commutative and cocommutative differential ASI bialgebras. The notion of the relative PCA Yang-Baxter equation (RPCA-YBE) in a relative PCA algebra is introduced, whose antisymmetric solutions give coboundary relative PCA bialgebras. The notions of $\mathcal{O}$-operators of relative PCA algebras and relative pre-PCA algebras are also introduced to give antisymmetric solutions of the RPCA-YBE.

The aim of this paper is to provide purely arithmetical characterisations of those natural numbers $n$ for which every non-degenerate set-theoretic solution of cardinality $n$ of the Yang--Baxter equation arising from a skew brace (sb-solution for short) satisfies some relevant properties, such as being a flip or being involutive. For example, it turns out that every sb-solution of cardinality $n$ has finite multipermutation level if and only if its prime factorisation $n= p_1^{\alpha_1} \ldots p_t^{\alpha_t}$ is cube-free, namely $\alpha_i\leq 2$ for every $i$, and $p_i$ does not divide $p_j^{\alpha_j}-1$ for $i\neq j$. Two novel constructions of skew braces will play a central role in our proofs. We shall also introduce the notion of supersoluble solution and show how this concept is related to that of supersoluble skew brace. In doing so, we have spotted an irreparable mistake in the proof of Theorem C [Ballester-Bolinches et al., Adv. Math. 455 (2024)], which characterizes soluble solutions in terms of soluble skew braces.

In this paper we present an encryption/decryption algorithm which use properties of finite MV-algebras, we proved that there are no commutative and unitary rings R such that Id (R) = L,where L is a finite BL-algebra which is not an MV-algebra and we give a method to generate BL-comets. Moreover, we give a final characterisation of finite BL-algebra and we proved that a finite BL-algebra is a comet or MV-algebras which are not chains.

We say that a formal deformation from an algebra $N$ to algebra $A$ is strongly flat if for every real number $e $ there is a real number $0<s<e$ such that this deformation specialised at $t=s$ gives an algebra isomorphic to $A$. We show that every strongly flat deformation from a finite-dimensional $C$-algebra $N$ to a semisimple $C$-algebra $A$ specialised at $t=s$ for all sufficiently small real numbers $s>0$ gives an algebra isomorphic to $A$. We also give a characterisation of semisimple algebras $A$ to which a given algebra $N$ cannot be deformed to. This gives a partial answer to a question of Michael Wemyss on Acons [14]. We also give a partial answer to question 6.5 from [1].

The adjunction between coalgebras and Hopf algebras, first described by Takeuchi, allows one to prove that the semi-abelian category of cocommutative Hopf algebras has enough $\mathcal E$-projective objects with respect to the class $\mathcal{E}$ of cleft extensions. One then proves that, for any cocommutative Hopf algebra, there exists a weak $\mathcal{E}$-universal normal (=central) extension. This fact allows one to apply the methods of categorical Galois theory to classify normal $\mathcal{E}$-extensions and to provide an explicit description of the fundamental group of a cocommutative Hopf algebra in terms of a generalized Hopf formula. Moreover, with any cleft extension, we associate a 5-term exact sequence in homology that can be seen as a Hopf-theoretic analogue of the classical Stallings-Stammbach exact sequence in group theory.

It is well-known that an averaging operator on a commutative associative algebra gives rise to a perm algebra. This paper lifts this process to the level of bialgebras. For this purpose, we first give an infinitesimal bialgebra structure for averaging commutative associative algebras and characterize it by double constructions of averaging Frobenius commutative algebras. To find the bialgebra counterpart of perm algebras that is induced by such averaging bialgebras, we need a new two-part splitting of the multiplication in a perm algebra, which differs from the usual splitting of the perm algebra (into the pre-perm algebra) by the characterized representation. This gives rise to the notion of an averaging-pre-perm algebra, or simply an apre-perm algebra. Furthermore, the notion of special apre-perm algebras which are apre-perm algebras with the second multiplications being commutative is introduced as the underlying algebra structure of perm algebras with nondegenerate symmetric left-invariant bilinear forms. The latter are also the induced structures of symmetric Frobenius commutative algebras with averaging operators. Consequently, a double construction of averaging Frobenius commutative algebra gives rise to a Manin triple of special apre-perm algebras. In terms of bialgebra structures, this means that an averaging commutative and cocommutative infinitesimal bialgebra gives rise to a special apre-perm bialgebra.

Motivated by recent advances in inverse semigroup theory, we investigate the growth of and identities satisfied by free left and free two-sided adequate monoids. We explicitly compute the growth of the monogenic free left adequate monoid with the usual unary monoid generating set and show it has intermediate growth owing to a connection with integer partitions. In the two-sided case, we establish a lower bound on the (idempotent) growth rate of the monogenic free adequate monoid, showing that it grows exponentially. We completely classify the enriched identities satisfied by the monogenic free left adequate monoid and deduce that it satisfies the same monoid identities as the sylvester monoid. In contrast, we show that the monogenic free two-sided adequate monoid satisfies no non-trivial monoid identities.

Let $\mathscr A$ be the Steenrod algebra over the field of characteristic two, $\mathbb F_2.$ Denote by $GL(q)$ the general linear group of rank $q$ over $\mathbb F_2.$ The algebraic transfer, introduced by W. Singer [Math. Z. 202 (1989), 493-523], is a rather effective tool for unraveling the intricate structure of the (mod-2) cohomology of the Steenrod algebra, ${\rm Ext}_{\mathscr A}^{q,*}(\mathbb F_2, \mathbb F_2).$ The Kameko homomorphism is one of the useful tools to study the dimension of the domain of the Singer transfer. Singer conjectured that the algebraic transfer is always a monomorphism, but this remains open for all homology degrees $q\geq 5.$ In this paper, by constructing a novel algorithm implemented in the computer algebra system OSCAR for computing $GL(q)$-invariants of the kernel of the Kameko homomorphism, we disprove Singer's conjecture for bidegree $(6, 6+36).$

In this paper we consider the algebra of upper triangular matrices UT$_n(F)$, endowed with a $\mathbb{Z}_2$-grading (superalgebra) and equipped with a superinvolution. These structures naturally arise in the context of Lie and Jordan superalgebras and play a central role in the theory of polynomial identities with involution, as showed in the framework developed by Aljadeff, Giambruno, and Karasik in [2]. We provide a complete description of the identities of UT$_4(F)$, where the grading is induced by the sequence $(0,1,0,1)$ and the superinvolution is the super-symplectic one. This work extends previous classifications obtained for the cases $n = 2$ and $n = 3$, and addresses an open problem for $n \geq 4$. In the last part of the paper, we investigate the image of multilinear polynomials on the superalgebra UT$_n(F)$ with superinvolution, showing that the image is a vector space if and only if $n \leq 3$, thus contributing to an analogue of the L'vov-Kaplansky conjecture in this context.

This paper is one of the series of papers which are dedicated to the complete classification of noncommutative conics. In this paper, we define and study noncommutative affine pencils of conics, and give a complete classification result. We also fully classify $4$-dimensional Frobenius algebras. It turns out that the classification of noncommutative affine pencils of conics is the same as the classification of $4$-dimensional Frobenius algebras.

Let $D$ be an integrally closed domain with quotient field $K$ and $A$ a torsion-free $D$-algebra that is finitely generated as a $D$-module and such that $A\cap K=D$. We give a complete classification of those $D$ and $A$ for which the ring $\text{Int}_K(A)=\{f\in K[X] \mid f(A)\subseteq A\}$ is a Pr\"ufer domain. If $D$ is a semiprimitive domain, then we prove that $\text{Int}_K(A)$ is Pr\"ufer if and only if $A$ is commutative and isomorphic to a finite direct product of almost Dedekind domains with finite residue fields, each of them satisfying a double-boundedness condition on its ramification indices and residue field degrees.

We investigate the annihilator condition $(a.c.)$ for skew Poincar\'e-Birkhoff-Witt extensions. We prove that some results about the annihilator condition $(a.c.)$ for skew polynomial rings also hold for skew PBW extensions. We also demonstrate that the behavior of annihilators is determined by the defining relations of the extension. Our results extend those corresponding presented for skew polynomial rings

We study the existence of formal Taylor expansions for functions defined on fields of generalised series. We prove a general result for the existence and convergence of those expansions for fields equipped with a derivation and an exponential function, and apply this to the case of standard fields of transseries, such as $\log$-$\exp$ transseries and $\omega$-series.

The symmetric inverse monoid $I_X$ on a set $X$ consists of all bijective functions whose domain and range are subsets of $X$ under the usual composition and inversion of partial functions. For an arbitrary infinite set $X$, we classify all maximal subsemigroups and maximal inverse subsemigroups of $I_X$ which contain the symmetric group Sym($X$) or any of the following subgroups of Sym($X$): the pointwise stabiliser of a finite subset of $X$, the stabiliser of an ultrafilter on $X$, or the stabiliser of a partition of $X$ into finitely many parts of equal cardinality.

Transposed Poisson algebra was introduced as a dual notion of the Poisson algebra by switching the roles played by the commutative associative operation and Lie operation in the Leibniz rule defining the Poisson algebra. Let $H$ be a Hopf algebra with a bijective antipode and $A$ an $H$-comodule transposed Poisson algebra. Assume that there exists an $H$-colinear map which is also an algebra map from $H$ to the transposed Poisson center of $A$. In this paper we generalize the fundamental theorem of $(A, H)$-Hopf modules to transposed Poisson $(A, H)$-Hopf modules and deduce relative projectivity in the category of transposed Poisson $(A, H)$-Hopf modules.