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In this paper, we carry out in an abstract order context some real subset combinatorial problems. Specifically, let [Formula: see text] be a finite poset, where [Formula: see text] is an order-reversing and involutive map such that [Formula: see text] for each [Formula: see text]. Let [Formula: see text] be the Boolean lattice with two elements and [Formula: see text] the family of all the order-preserving 2-valued maps [Formula: see text] such that [Formula: see text] if [Formula: see text] for all [Formula: see text]. In this paper, we build a family [Formula: see text] of particular subsets of [Formula: see text], that we call [Formula: see text]-bases on [Formula: see text], and we determine a bijection between the family [Formula: see text] and the family [Formula: see text]. In such a bijection, a [Formula: see text]-basis [Formula: see text] on [Formula: see text] corresponds to a map [Formula: see text] whose restriction of [Formula: see text] to [Formula: see text] is the smallest 2-valued partial map on [Formula: see text] which has [Formula: see text] as its unique extension in [Formula: see text]. Next we show how each [Formula: see text]-basis on [Formula: see text] becomes, in a particular context, a sub-system of a larger system of linear inequalities, whose compatibility implies the compatibility of the whole system.

This paper treats the strict semi-stability of the symmetric powers [Formula: see text] of a stable vector bundle [Formula: see text] of rank [Formula: see text] with even degree on a smooth projective curve [Formula: see text] of genus [Formula: see text]. The strict semi-stability of [Formula: see text] is equivalent to the orthogonality of [Formula: see text] or the existence of a bisection on the ruled surface [Formula: see text] whose self-intersection number is zero. A relation between the two interpretations is investigated in this paper through elementary transformations. This paper also gives a classification of [Formula: see text] with strictly semi-stable [Formula: see text]. Moreover, it is shown that when [Formula: see text] is stable, every symmetric power [Formula: see text] is stable for all but a finite number of [Formula: see text] in the moduli of stable vector bundles of rank [Formula: see text] with fixed determinant of even degree on [Formula: see text].

In this paper, we introduce and study a new kind of graph related to a unitary module [Formula: see text] on a commutative ring [Formula: see text] with identity, namely the annihilators comaximal graph of submodules of [Formula: see text], denoted by [Formula: see text]. The (undirected) graph [Formula: see text] is with vertices of all non-trivial submodules of [Formula: see text] and two vertices [Formula: see text] of [Formula: see text] are adjacent if and only if their annihilators are comaximal ideals of [Formula: see text], i.e. [Formula: see text]. The main purpose of this paper is to investigate the interplay between the graph-theoretic properties of [Formula: see text] and the module-theoretic properties of [Formula: see text]. We study the annihilators comaximal graph [Formula: see text] in terms of the powers of the decomposition of [Formula: see text] to product distinct prime numbers in some special cases.

Let R be a non-commutative ring, and I be an ideal of R. In this paper, we generalize the definition of the zero-divisor graph of R with respect to I, and define several generalized zero-divisor graphs of R with respect to I. In this paper, we investigate the ring-theoretic properties of R and the graph-theoretic properties of all the generalized zero-divisor graphs. We study some basic properties of these generalized zero-divisor graphs related to the connectedness, the diameter and the girth. We also investigate some properties of these generalized zero-divisor graphs with respect to primal ideals.

This is about the paper “Bi-interior ideals of semigroups” by M. Murali Krishna Rao in Discuss. Math. Gen. Algebra Appl. 38 (2018) 69–78. According to Theorem 3.11 (also Theorem 3.3(8)) of the paper, the intersection of a bi-interior ideal [Formula: see text] of a semigroup [Formula: see text] and a subsemigroup [Formula: see text] of [Formula: see text] is a bi-interior ideal of [Formula: see text]. Regarding to Theorem 3.6, every bi-interior ideal of a regular semigroup is an ideal of [Formula: see text]. We give an example that the above two results are not true for semigroups. According to the same paper, if [Formula: see text] is a regular semigroup then, for every bi-interior ideal [Formula: see text], every ideal [Formula: see text] and every left ideal [Formula: see text] of [Formula: see text], we have [Formula: see text]. The proof is wrong, we provide the corrected proof. In most of the results of the paper the assumption of unity is not necessary. Care should be taken about the proofs in the paper.

Let [Formula: see text] be a field of characteristic zero and [Formula: see text] the algebra of [Formula: see text] matrices over [Formula: see text]. By the classical Amitsur–Levitzki theorem, it is well known that [Formula: see text] is the smallest degree of a standard polynomial identity of [Formula: see text]. A theorem due to Rowen shows that when the symplectic involution [Formula: see text] is considered, the standard polynomial of degree [Formula: see text] in symmetric variables is an identity of [Formula: see text]. This means that when only certain kinds of matrices are considered in the substitutions, the minimal degree of a standard identity may not remain being the same. In this paper, we present some results about the minimal degree of standard identities in skew or symmetric variables of odd degree of [Formula: see text] in the symplectic graded involution case. Along the way, we also present the minimal total degree of a double Capelli polynomial identity in symmetric variables of [Formula: see text] with transpose involution.

Let [Formula: see text] be a finite group. How minimal subgroups can be embedded in [Formula: see text] is a question of particular interest in studying the structure of [Formula: see text]. A subgroup [Formula: see text] of [Formula: see text] is called [Formula: see text]-permutable in [Formula: see text] if [Formula: see text] for all Sylow subgroups [Formula: see text] of [Formula: see text]. A subgroup [Formula: see text] of [Formula: see text] is called [Formula: see text]-embedded in [Formula: see text] if there exists a normal subgroup [Formula: see text] of [Formula: see text] such that [Formula: see text] and [Formula: see text], where [Formula: see text] is the subgroup of [Formula: see text] generated by all those subgroups of [Formula: see text] which are [Formula: see text]-permutable in [Formula: see text]. In this paper, we investigate the structure of the finite group [Formula: see text] with [Formula: see text]-embedded subgroups.

Let [Formula: see text] be a graph with edge set [Formula: see text]. For an edge [Formula: see text] in [Formula: see text], we define [Formula: see text], where [Formula: see text] and [Formula: see text] are degrees of vertices [Formula: see text] and [Formula: see text] in [Formula: see text], respectively. For [Formula: see text], the graph invariants [Formula: see text], [Formula: see text] and [Formula: see text] are defined as [Formula: see text], [Formula: see text] and [Formula: see text], where [Formula: see text] means that the edges [Formula: see text] and [Formula: see text] are incident. In this paper, some relationship between these graph invariants and some classical topological indices were presented. Moreover, some bounds for [Formula: see text], [Formula: see text] and [Formula: see text] are obtained and trees with the first through the third smallest [Formula: see text] and [Formula: see text], as well as the trees with the first through the forth smallest [Formula: see text] are also characterized.

Let [Formula: see text] be a lattice, [Formula: see text] an ideal of [Formula: see text] and [Formula: see text] the set of all elements of [Formula: see text] which are not prime to [Formula: see text]. In this paper, we investigate some interesting properties of [Formula: see text] and introduce the total graph of a lattice [Formula: see text] with respect to an ideal [Formula: see text]. It is the graph with all elements of [Formula: see text] as vertices and for distinct [Formula: see text], the vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. We investigated the basic properties of this graph and its induced subgraphs as diameter, girth, clique number, cut vertex and independence number.

In this paper, we prove that if [Formula: see text] is a partitioning ideal of an [Formula: see text]-semiring [Formula: see text] with respect to two subsets [Formula: see text] and [Formula: see text] of [Formula: see text], then the quotient [Formula: see text]-semirings [Formula: see text] and [Formula: see text] are isomorphic. Also, the concepts of maximal homomorphism, steady homomorphism, one-one homomorphism and [Formula: see text]-homomorphism are generalized for [Formula: see text]-semirings and obtained relation between them. Finally, the fundamental theorem of semiring homomorphism is generalized for [Formula: see text]-semirings.

The irregularity of a simple undirected graph G is defined as irr(G) = Sigma(uv is an element of E(G)) vertical bar d(G)(u) - d(G) (v) vertical bar, where d(G) (u) denotes the degree of a vertex u is an element of V (G). The total irregularity of a graph as a new measure of graph irregularity is defined as irr(t) (G) = 1/2 Sigma(u,v is an element of V (G) )vertical bar d(G) (u) - d(G) (v) vertical bar. In this paper, the irregularity and the total irregularity of fractal graphs and the derived graphs from a class of fractal graphs are investigated. Exact formulae are presented for the computation of the irregularity and total irregularity of fractal-type graphs in terms of the parameters of the underlying graphs.

This paper studies the bi-compact-[Formula: see text]-open topology on the family [Formula: see text] of all continuous functions from a completely regular space [Formula: see text] to the space [Formula: see text]. We compare this topology with open-point topology, bi-point-open topology and bi-compact-open topology and mention an example of a space asked in [P. Garg and S. Kundu, The compact-[Formula: see text]-open topology on [Formula: see text], Topology Appl. 159(8) (2012) 2082–2089] which satisfies [Formula: see text]. We also investigate submertrizability, mertrizability, separability, [Formula: see text]ech-completeness and Baire property of bi-compact-[Formula: see text]-open topology on [Formula: see text].

Denote $M_k$ the set of complex $k$ by $k$ matrices. We will analyze here quantum channels $\phi_L$ of the following kind: given a measurable function $L:M_k\to M_k$ and the measure $\mu$ on $M_k$ we define the linear operator $\phi_L:M_k \to M_k$, via the expression $\rho \,\to\,\phi_L(\rho) = \int_{M_k} L(v) \rho {L(v)}^\dagger \, \dm(v)$. A recent paper by T. Benoist, M. Fraas, Y. Pautrat, and C. Pellegrini is our starting point. They considered the case where $L$ was the identity. Under some mild assumptions on the quantum channel $\phi_L$ we analyze the eigenvalue property for $\phi_L$ and we define entropy for such channel. For a fixed $\mu$ (the \textit{a priori} measure) and for a given a Hamiltonian $H: M_k \to M_k$ we present a version of the Ruelle Theorem: a variational principle of pressure (associated to such $H$) related to an eigenvalue problem for the Ruelle operator. We introduce the concept of Gibbs channel. We also show that for a fixed $\mu$ (with more than one point in the support) the set of $L$ such that it is $\phi$-Erg (also irreducible) for $\mu$ is a generic set. We describe a related process $X_n$, $n\in \mathbb{N}$, taking values on the projective space $ P(\C^k)$ and analyze the question of the existence of invariant probabilities. We also consider an associated process $\rho_n$, $n\in \mathbb{N}$, with values on $\mathcal{D}_k$ ($\mathcal{D}_k$ is the set of density operators). Via the barycenter we associate the invariant probabilities mentioned above with the density operator which is fixed for $\phi_L$., Comment: We correct some mistakes in some proofs and we add a new section on genericty of ergodicity. We change the title

Let [Formula: see text] be a minimal parabolic subgroup of a real reductive Lie group [Formula: see text] and [Formula: see text] a closed subgroup of [Formula: see text]. Then it is proved by Kobayashi and Oshima that the regular representation [Formula: see text] contains each irreducible representation of [Formula: see text] at most finitely many times if the number of [Formula: see text]-orbits on [Formula: see text] is finite. Moreover, they also proved that the multiplicities are uniformly bounded if the number of [Formula: see text]-orbits on [Formula: see text] is finite, where [Formula: see text] are complexifications of [Formula: see text], respectively, and [Formula: see text] is a Borel subgroup of [Formula: see text]. In this paper, we prove that the multiplicities of the representations of [Formula: see text] induced from a parabolic subgroup [Formula: see text] in the regular representation on [Formula: see text] are uniformly bounded if the number of [Formula: see text]-orbits on [Formula: see text] is finite. For the proof of this claim, we also show the uniform boundedness of the dimensions of the spaces of group invariant hyperfunctions using the theory of holonomic [Formula: see text]-modules.

The Young type inequality asserts that if [Formula: see text] are two positive real numbers, then for each [Formula: see text], we have [Formula: see text] In this paper, we obtain some new inequalities using two different weights. For example, if [Formula: see text], then [Formula: see text] [Formula: see text] where [Formula: see text] In addition, we refine some matrix inequalities for Unitarily invariant norms by applying the deduced numerical inequalities.

Let [Formula: see text] be a bounded [Formula: see text]-hyperconvex domain ([Formula: see text]) and [Formula: see text] be a weighted function. In this paper, we first prove a characterization of the finite weighted energy class [Formula: see text] and give an upper-bound for [Formula: see text]-capacity in the class [Formula: see text]. Then, by using these estimates, we construct some classes of finite weighted energy functions in [Formula: see text]-subharmonic functions class and give some corollaries about the relationship between these finite weighted energy functions classes.

In this paper, we define a module [Formula: see text] to be [Formula: see text] if and only if intersection of each pair of [Formula: see text]-closed direct summands is also a direct summand of [Formula: see text]. We investigate structural properties of [Formula: see text]-modules and locate the implications between the other module properties which are essentially based on direct summands. We deal with decomposition theory as well as direct summands of [Formula: see text]-modules. We apply our results to matrix rings. To this end, it is obtained that the [Formula: see text] property is not Morita invariant.

In this paper, we introduce Fibonacci backward difference operator [Formula: see text] of fractional order l by the composition of Fibonacci band matrix [Formula: see text] and difference operator [Formula: see text] of fractional order l, defined by [Formula: see text] and introduce sequence spaces [Formula: see text] and [Formula: see text] We present some topological properties, obtain Schauder basis and determine [Formula: see text]-, [Formula: see text]- and [Formula: see text]-duals of the spaces [Formula: see text] and [Formula: see text] We characterize certain classes of matrix mappings from the spaces [Formula: see text] and [Formula: see text] to any of the space [Formula: see text] [Formula: see text] [Formula: see text] or [Formula: see text] Finally we compute necessary and sufficient conditions for a matrix operator to be compact on the spaces [Formula: see text] and [Formula: see text]

In this paper, we study characterizations of Griffiths semi-positivity through [Formula: see text]-estimates of the [Formula: see text]-equation and [Formula: see text]-extension theorems for symmetric powers of a holomorphic vector bundle. We also investigate several versions of the converse of the Demailly–Skoda theorem.

An r-dynamic coloring of a graph G is a proper coloring c of the vertices such that [Formula: see text] for each vertex [Formula: see text]. The r-dynamic chromatic number of a graph G is the minimum k such that G has an r-dynamic coloring with k colors. In this paper, we obtain the r-dynamic chromatic number of brick product graphs.

Let [Formula: see text] be a commutative ring with nonzero identity. The comaximal graph of [Formula: see text], denoted by [Formula: see text], is a simple graph with vertex set [Formula: see text], and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. Let [Formula: see text] be an induced subgraph of [Formula: see text] with nonunit elements of [Formula: see text] as vertices. In this paper, we describe the normalized Laplacian spectrum of [Formula: see text], and we determine it for some values of [Formula: see text], where [Formula: see text] is the ring of integers modulo [Formula: see text]. Moreover, we investigate the normalized Laplacian energy and general Randic index of [Formula: see text].

In this paper, we deal with the periodicity of solutions of the following general system rational of difference equations: [Formula: see text] where [Formula: see text] [Formula: see text] and the initial conditions are arbitrary nonzero real numbers.

The distance signless Laplacian eigenvalues [Formula: see text] of a connected graph [Formula: see text] are the eigenvalues of the distance signless Laplacian matrix of [Formula: see text], defined as [Formula: see text], where [Formula: see text] is the distance matrix of [Formula: see text] and [Formula: see text] is the diagonal matrix of vertex transmissions of [Formula: see text]. In this paper, we define and investigate the distance signless Laplacian Estrada index of a graph [Formula: see text] as [Formula: see text], and obtain some upper and lower bounds for [Formula: see text] in terms of other graph invariants. We also obtain some relations between [Formula: see text] and the auxiliary distance signless Laplacian energy of [Formula: see text].

In this paper, we give the characterization of the unit groups of semisimple group algebras of some non-metabelian groups of order [Formula: see text]. This study completes the study of unit groups of semisimple group algebras of all groups up to order [Formula: see text], except that of the symmetric group [Formula: see text] and groups of order [Formula: see text].

The [Formula: see text]-index of a graph [Formula: see text], denoted by [Formula: see text], is defined as the sum of weights [Formula: see text] over all edges [Formula: see text] of [Formula: see text], where [Formula: see text] denotes the degree of a vertex [Formula: see text]. In this paper, we give sharp upper bounds of the [Formula: see text]-index (forgotten topological index) over bicyclic graphs, in terms of the order and maximum degree.

In [H. Rashidi, A. Golchin and H. Mohammadzadeh saany, On [Formula: see text]-flat acts, Categ. Gen. Algebr. Struct. Appl. 12(1) (2020) 175–197], the study of [Formula: see text]-flatness property of right acts [Formula: see text] over a monoid [Formula: see text] that can be described by means of when the functor [Formula: see text]-preserves some pullbacks is initiated. In this paper, we extend these results to [Formula: see text]-posets and present equivalent description of [Formula: see text]-po-flatness of [Formula: see text]-posets. We show that [Formula: see text]-flatness does not imply torsion freeness in [Formula: see text]-posets and give some general properties and a characterization of pomonoids for which some other properties of their posets imply this condition.

In this paper, we consider finite alternating sums derived from the generalized Fibonacci numbers [Formula: see text] [Formula: see text] and [Formula: see text], where [Formula: see text] and [Formula: see text] are positive integers with [Formula: see text], [Formula: see text]. Applying the greatest integer function to these sums, we obtain some equalities involving the generalized Fibonacci numbers.

The large rank of a finite semigroup [Formula: see text] is the least number [Formula: see text] such that every subset of [Formula: see text] with [Formula: see text] elements generates [Formula: see text]. This paper obtains the large ranks of [Formula: see text] and [Formula: see text], the semigroups of singular transformations, injective partial and partial transformations on a finite chain [Formula: see text], which preserve or reverse the order, respectively. As a consequence, we obtain the large ranks of [Formula: see text] and [Formula: see text], the semigroups of injective order-preserving partial transformations and order-preserving partial transformations on [Formula: see text], respectively.

Let [Formula: see text] and [Formula: see text] be two Lie algebras over an arbitrary field [Formula: see text], and let [Formula: see text] be the semidirect sum of [Formula: see text] by [Formula: see text]. In this paper, we give the structure of derivation algebra of [Formula: see text]; then as a consequence we illustrate the structure and dimension derivation algebra of Heisenberg Lie algebras.

In this paper, we study the existence of multiple solutions for the boundary value problem [Formula: see text] where [Formula: see text] is a bounded domain with smooth boundary in [Formula: see text], nonlinearity [Formula: see text] is subcritical growth and does not satisfy the Ambrosetti–Rabinowitz (AR) condition and [Formula: see text] is the subelliptic operator of the type [Formula: see text] We use variational methods (Mountain pass theorem and Ekeland variational principle).

A module [Formula: see text] is called [Formula: see text]-separable if every proper finitely generated submodule of [Formula: see text] is contained in a proper finitely generated direct summand of [Formula: see text]. Indecomposable [Formula: see text]-separable modules are shown to be exactly the simple modules. While direct summands of an [Formula: see text]-separable module do not inherit the property, in general, the question of the stability under direct sums is unanswered. But we obtain some partial answers. It is shown that any infinite direct sum of [Formula: see text]-separable modules is [Formula: see text]-separable. Also, we prove that if [Formula: see text] and [Formula: see text] are [Formula: see text]-separable modules such that [Formula: see text] is [Formula: see text]-projective, then [Formula: see text] is [Formula: see text]-separable. We conclude the paper by providing some characterizations of several classes of rings in terms of [Formula: see text]-separable modules. Among others, we prove that the class of rings [Formula: see text] for which every (injective) [Formula: see text]-module is [Formula: see text]-separable is exactly that of semisimple rings.