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The enhanced power graph P E (S) of a semigroup S is a simple graph whose vertex set is S and two vertices x , y ∈ S are adjacent if and only if x , y ∈ ⟨ z ⟩ for some z ∈ S , where ⟨ z ⟩ is the subsemigroup generated by z. In this paper, we first describe the structure of P E (S) for an arbitrary semigroup S , and then discuss the connectedness of P E (S). Further, we characterize the semigroup S in the cases when P E (S) is separately a complete, bipartite, regular, tree and null graph. The planarity, together with the minimum degree and independence number, of P E (S) is also investigated. The chromatic number of a spanning subgraph, i.e., the cyclic graph, of P E (S) is proved to be countable. In the final part of this paper, we construct an example of a semigroup S such that the chromatic number of P E (S) need not be countable. [ABSTRACT FROM AUTHOR]

A simple undirected regular graph is said to be semisymmetric if it is edge-transitive but not vertex-transitive. For a semisymmetric graph Γ of order 2 p 3 , p a prime, it is well known that Γ is bipartite with two biparts having equal size. The complete classification of such graphs has been given for the full automorphism group Aut (Γ) acting unfaithfully on at least one bipart of Γ , which shows that there is only one infinite family of such graphs with valency p 2. The graphs of this kind have been determined when Aut (Γ) acts faithfully and primitively on at least one bipart of Γ , and thus there is only one remaining case for classifying such graphs of valency p 2 , Aut (Γ) acting faithfully and imprimitively on both biparts of Γ , which is dealt with in this paper. As a result, there is only one infinite family of semisymmetric graphs of order 2 p 3 with valency p 2. [ABSTRACT FROM AUTHOR]

In this paper we continue the study of various ring theoretic properties of Morita contexts. Necessary and sufficient conditions are obtained for a general Morita context or a trivial Morita context or a formal triangular matrix ring to satisfy a certain ring property which is among being Kasch, completely primary, quasi-duo, 2-primal, NI, semiprimitive, projective-free, etc. We also characterize when a general Morita context is weakly principally quasi-Baer or strongly right mininjective. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the electrical Lie algebra e D 5 is isomorphic to the semidirect product of s p 4 and a 2-step nilpotent Lie algebra. Furthermore, we classify the irreducible highest weight modules for e D 5 . [ABSTRACT FROM AUTHOR]

The metric dimension problem is called navigation problem due to its application to robot navigation in space. Further this concept has wide applications in motion planning, sonar and loran station, and so on. In this paper, we study certain results on the metric dimension, upper dimension and resolving number of extended annihilating-ideal graph E A G (R) associated to a commutative ring R , denoted by dim M (E A G (R)) , dim + (E A G (R)) and res (E A G (R)) , respectively. Here we prove the finiteness conditions of dim M (E A G (R)) and dim + (E A G (R)). In addition, we characterize dim M (E A G (R)) , dim + (E A G (R)) and res (E A G (R)) for artinian rings and the direct product of rings. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of a product structure on a 3-Bihom-Lie algebra, which is a Nijenhuis operator with some conditions. We prove that a 3-Bihom-Lie algebra has a product structure if and only if it is the direct sum of two vector spaces which are also Bihom-subalgebras. Then we give four special conditions under each of which a 3-Bihom-Lie algebra has a special decomposition. Similarly, we introduce a complex structure on a 3-Bihom-Lie algebra and there are also four types of special complex structures. Finally, we establish the relation between a complex structure and a product structure. [ABSTRACT FROM AUTHOR]

In this paper, we prove a reduction result on wide subcategories of abelian categories which is similar to the Calabi-Yau reduction, silting reduction and τ -tilting reduction. More precisely, if an abelian category A admits a recollement relative to abelian categories A ′ and A ′ ′ , which is denoted by (A ′ , A , A ′ ′ , i ∗ , i ∗ , i ! , j ! , j ∗ , j ∗) , then the assignment C ↦ j ∗ (C) defines a bijection between wide subcategories in A containing i ∗ (A ′) and wide subcategories in A ′ ′. Moreover, a wide subcategory C of A containing i ∗ (A ′) admits a new recollement relative to A ′ and j ∗ (C) which is induced from the original recollement. [ABSTRACT FROM AUTHOR]

Let (A , B , C) be a recollement of extriangulated categories. In this paper we introduce the global dimension and extension dimension of extriangulated categories, and give some upper bounds of global dimensions and extension dimensions of the categories involved in (A , B , C) , which give a simultaneous generalization of some results in recollements of abelian categories and triangulated categories. [ABSTRACT FROM AUTHOR]

Let G n ([ − 1 ] i) denote the set of all connected graphs on n vertices having distance eigenvalue − 1 of multiplicity i. By using the distribution of the third largest distance eigenvalue and the second least distance eigenvalue of a connected graph, in this paper we completely characterize the graphs in G n ([ − 1 ] i) , where i = n − 1 , n − 2 , n − 3 or n − 4. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity, M be an R -module, L (M) denote the set of all submodules of M and G ⊆ L (M) \ { 0 M }. For any submodule N of M , we set G V d (N) = { K ∈ G : K ⊆ N } and G ζ d (M) = { G V d (N) : N ∈ L (M) }. Consider χ ⊆ L (R) \ { R } , where L (R) is the set of all ideals of R. We set χ V (I) = { J ∈ χ : I ⊆ J } and χ ζ (R) = { χ V (I) : I ∈ L (R) } for any ideal I of R. In this paper, we investigate when, for arbitrary χ and G as above, χ ζ (R) and G ζ d (M) form a topology and a semimodule, respectively. We investigate the structure of G ζ d (M) in the case that it is a semimodule. [ABSTRACT FROM AUTHOR]

The spectra of generalized Cayley graphs of finite abelian groups are investigated in this paper. For a generalized Cayley graph X of a finite group G , the canonical double covering of X is the direct product X × K 2. In this paper, integral generalized Cayley graphs on finite abelian groups are characterized, using the characterization of the spectra of integral Cayley graphs. As an application, the integral generalized Cayley graphs on Z p × Z q and Z 2 n are investigated, where p and q are odd prime numbers. [ABSTRACT FROM AUTHOR]

A signed graph G ˙ = (G , σ) is a graph G = (V (G) , E (G)) with vertex set V (G) and edge set E (G) , together with a function σ : E → { + 1 , − 1 } assigning a positive or negative sign to each edge. In this paper, we present a more elementary proof for the matrix-tree theorem of signed graphs, which is based on the relations between the incidence matrices and the Laplcians of signed graphs. As an application, we also obtain the results of Monfared and Mallik about the matrix-tree theorem of graphs for signless Laplacians. [ABSTRACT FROM AUTHOR]

For a ring R (not necessarily commutative) with identity, the comaximal graph of R , denoted by Ω (R) , is a graph whose vertices are all the nonunit elements of R , and two distinct vertices a and b are adjacent if and only if R a + R b = R. In this paper we consider a subgraph Ω 1 (R) of Ω (R) induced by R \ U ℓ (R) , where U ℓ (R) is the set of all left-invertible elements of R. We characterize those rings R for which Ω 1 (R) \ J (R) is a complete graph or a star graph, where J (R) is the Jacobson radical of R. We investigate the clique number and the chromatic number of the graph Ω 1 (R) \ J (R) , and we prove that if every left ideal of R is symmetric, then this graph is connected and its diameter is at most 3. Moreover, we completely characterize the diameter of Ω 1 (R) \ J (R). We also investigate the properties of R when Ω 1 (R) is a split graph. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with nonzero identity and n be a positive integer. In this paper, we introduce and investigate a new subclass of ϕ - n -absorbing primary ideals, which are called ϕ - (n , N) -ideals. Let ϕ : I (R) → I (R) ∪ { ∅ } be a function, where I (R) denotes the set of all ideals of R. A proper ideal I of R is called a ϕ - (n , N) -ideal if x 1 ⋯ x n + 1 ∈ I \ ϕ (R) and x 1 ⋯ x n ∉ I imply that the product of x n + 1 with (n − 1) of x 1 , ... , x n is in 0 for all x 1 , ... , x n + 1 ∈ R. In addition to giving many properties of ϕ - (n , N) -ideals, we also use the concept of ϕ - (n , N) -ideals to characterize rings that have only finitely many minimal prime ideals. [ABSTRACT FROM AUTHOR]

The inclusion ideal graph I n (S) of a semigroup S is an undirected simple graph whose vertices are all the nontrivial left ideals of S and two distinct left ideals I , J are adjacent if and only if either I ⊂ J or J ⊂ I. The purpose of this paper is to study algebraic properties of the semigroup S as well as graph theoretic properties of I n (S). We investigate the connectedness of I n (S) and show that the diameter of I n (S) is at most 3 if it is connected. We also obtain a necessary and sufficient condition of S such that the clique number of I n (S) is the number of minimal left ideals of S. Further, various graph invariants of I n (S) , viz. perfectness, planarity, girth, etc., are discussed. For a completely simple semigroup S , we investigate properties of I n (S) including its independence number and matching number. Finally, we obtain the automorphism group of I n (S). [ABSTRACT FROM AUTHOR]

Let H be a finite-dimensional pointed rank one Hopf algebra of nilpotent type over a finite group G. In this paper, we investigate the McKay matrix W V of H for tensoring with the 2-dimensional indecomposable H -module V := M (2 , 0). It turns out that the characteristic polynomial, eigenvalues and eigenvectors of W V are related to the character table of the finite group G and a kind of generalized Fibonacci polynomial. Moreover, we construct some eigenvectors of each eigenvalue for W V by using the factorization of the generalized Fibonacci polynomial. As an example, we explicitly compute the characteristic polynomial and eigenvalues of W V and give all eigenvectors of each eigenvalue for W V when G is a dihedral group of order 4 N + 2. [ABSTRACT FROM AUTHOR]

In this paper, we provide a diagrammatic approach to study the branching rules for cell modules on a sequence of walled Brauer algebras. This approach also allows us to calculate the structure constants of multiplication over the Grothendieck ring of the sequence. [ABSTRACT FROM AUTHOR]

Suppose that L is a Lie superalgebra over a field F of characteristic different from 2 and 3. In this paper, the so-called 5-sequence of cohomology for a central extension of L is constructed and is proved to be exact. Moreover, the multiplier of L is proved to be isomorphic to the second cohomology group with coefficients in the trivial module of L. Finally, an upper bound of the superdimension of the second cohomology group is given in the situation when L is nilpotent and finite-dimensional. [ABSTRACT FROM AUTHOR]

An abelian group A is called a T I -group if every associative ring with the additive group A is filial. The filiality of a ring R means that the ring R is associative and all ideals of any ideal of R are ideals in R. In this paper, torsion-free T I -groups are described up to the structure of associative nil groups. It is also proved that, for torsion-free abelian groups that are not associative nil, the condition T I implies the indecomposability and homogeneity. The paper contains constructions of 2 ℵ 0 such groups of any rank from 2 to 2 ℵ 0 which are pairwise non-isomorphic. [ABSTRACT FROM AUTHOR]

In this paper we present two new families of 2- (v , k , λ) designs with a flag-transitive and point-primitive automorphism group of product action type. More surprisingly, one of them is still a family of 2- (v , k , λ) designs with a flag-transitive and point-imprimitive automorphism group. [ABSTRACT FROM AUTHOR]

The distance matrix of a connected graph G , denoted by D (G) , is the matrix whose rows and columns are indexed by the vertex set V (G) such that the (v i , v j) -entry is d (v i , v j) , where v i , v j ∈ V (G). The distance signature sig (D (G)) of G is the inertia of D (G). In this paper, we determine the distance signature of the extended (co-extended) incidence graph of an affine design. Furthermore, we state that an open Graffiti conjecture is true for the extended (co-extended) incidence graphs of affine designs by investigating the lower bound of the matching number. [ABSTRACT FROM AUTHOR]

In this paper, we describe multiplicative derivations on the set of all rank- s matrices of M n (K) over a field K with a relatively small integer s. Concretely, for fixed integers n , s satisfying 1 ≤ s ≤ n / 2 and n ≥ 2 , we prove that if a map δ : M n (K) → M n (K) satisfies δ (x y) = δ (x) y + x δ (y) for any two rank- s matrices x , y ∈ M n (K) , then there exists a derivation D of M n (K) such that δ (x) = D (x) for each rank- k matrix x ∈ M n (K) with 0 ≤ k ≤ s. As an application, we prove that a multiplicative derivation on a special subset of M n (K) must be a derivation. [ABSTRACT FROM AUTHOR]

A finite p -group G is called an A t -group if t is the minimal non-negative integer such that all subgroups of index p t of G are abelian. The finite p -groups G with H ′ = G ′ for all A 2 -subgroups H of G are classified completely in this paper. As an application, a problem proposed by Berkovich is solved. [ABSTRACT FROM AUTHOR]

Let L be a simple modular Lie superealgebra of Witt type or special type over an algebraically closed field of characteristic p > 2. In this paper, all the superbiderivations of L are studied. By means of weight space decompositions with respect to a suitable torus and the standard Z -grading structures of L , we show that the supersymmetric superbiderivations of L are zero. Generalizing a result on the skewsymmetric biderivation of Lie algebras to the super case, we find that all the superbiderivations of L are inner. As applications, the linear supercommuting maps and the supercommutative post-Lie superalgebra structures on L are described. [ABSTRACT FROM AUTHOR]

The spectra of generalized Cayley graphs of finite abelian groups are investigated in this paper. For a generalized Cayley graph X of a finite group G , the canonical double covering of X is the direct product X × K 2. In this paper, integral generalized Cayley graphs on finite abelian groups are characterized, using the characterization of the spectra of integral Cayley graphs. As an application, the integral generalized Cayley graphs on Z p × Z q and Z 2 n are investigated, where p and q are odd prime numbers. [ABSTRACT FROM AUTHOR]

Let A be a completely decomposable homogeneous torsion-free abelian group of rank n (n ≥ 2). Let Θ m (n) = A ⋊ ⟨ α ⟩ be the split extension of A by an automorphism α which is a cyclic permutation of the direct components twisted by a rational integer m. Then Θ m (n) is an infinite soluble group. In this paper, the residual finiteness of Θ m (n) is investigated. [ABSTRACT FROM AUTHOR]

A set { a 1 , a 2 , ... , a m } of positive integers is called a Diophantine m -tuple if a i a j + 1 is a perfect square for all 1 ≤ i < j ≤ m. Let { a , b , c } be the Diophantine triple with c > max { a , b }. In this paper, we find the condition for the reduction of third element c , and using this result, we prove the extendibility of Diophantine pair { F k − 1 F k + 1 , F k − 2 F k + 2 } , where F n is the n -th Fibonacci number. [ABSTRACT FROM AUTHOR]

Let p be an odd prime, and n , k be nonnegative integers. Let D n , k (1 , x) be the reversed Dickson polynomial of the (k + 1) -th kind. In this paper, by using Hermite's criterion, we study the permutational properties of the reversed Dickson polynomials D n , k (1 , x) over finite fields in the case of n = m p s with 0 < m < p − 1. In particular, we provide some precise characterizations for D n , k (1 , x) being permutation polynomials over finite fields with characteristic p when n = 2 p s , or n = 3 p s , or n = 4 p s. [ABSTRACT FROM AUTHOR]

Jordan D-bialgebras were introduced by Zhelyabin. In this paper, we use a new approach to study Jordan D-bialgebras by a new notion of the dual representation of the regular representation of a Jordan algebra. Motivated by the essential connection between Lie bialgebras and Manin triples, we give an explicit proof of the equivalence between Jordan D-bialgebras and a class of special Jordan–Manin triples called double constructions of pseudo-euclidean Jordan algebras. We also show that a Jordan D-bialgebra leads to the Jordan Yang–Baxter equation under the coboundary condition and an antisymmetric nondegenerate solution of the Jordan Yang–Baxter equation corresponds to an antisymmetric bilinear form, which we call a Jordan symplectic form on Jordan algebras. Furthermore, there exists a new algebra structure called pre-Jordan algebra on Jordan algebras with a Jordan symplectic form. [ABSTRACT FROM AUTHOR]

This paper introduces the notion of depth with respect to ideals for unbounded DG-modules, and gives a reduction formula and the local nature of this depth. As applications, we provide several bounds of the depth in special cases, and recover and generalize the known results about the depth of complexes. In addition, the width with respect to ideals for unbounded DG-modules is investigated and the depth and width formulas for DG-modules are generalized. [ABSTRACT FROM AUTHOR]

Let p be a prime. In this paper, a complete classification of edge-transitive N -covers of a cubic symmetric graph of order 2 p is given for the case when N is a two-generator 2-group whose derived subgroup is either isomorphic to Z 2 3 or generated by at most two elements. As an application, it is shown that 11 is the smallest value of n for which there exist infinitely many cubic semisymmetric graphs with order of the form 2 n p. [ABSTRACT FROM AUTHOR]

Let g be a finite dimensional complex simple Lie algebra with Cartan subalgebra h. Then C [ h ] has a g -module structure if and only if g is of type A or of type C ; this is called the polynomial module of rank one. In the quantum version, the rank one polynomial modules over U q (sl 2) have been classified. In this paper, we prove that the quantum group U q (sl 3) has no rank one polynomial module. [ABSTRACT FROM AUTHOR]

Let Δ be a simplicial complex on [ n ]. The NF -complex of Δ is the simplicial complex δ NF (Δ) on [ n ] for which the facet ideal of Δ is equal to the Stanley–Reisner ideal of δ NF (Δ). Furthermore, for each k = 2 , 3 , ... , we introduce the k th NF -complex δ NF (k) (Δ) , which is inductively defined as δ NF (k) (Δ) = δ NF δ NF (k − 1) (Δ) by setting δ NF (1) (Δ) = δ NF (Δ). One can set δ NF (0) (Δ) = Δ. The NF -number of Δ is the smallest integer k > 0 for which δ NF (k) (Δ) ≃ Δ. In the present paper we are especially interested in the NF -number of a finite graph, which can be regraded as a simplicial complex of dimension one. It is shown that the NF -number of the finite graph K n ⊔ K m on [ n + m ] , which is the disjoint union of the complete graphs K n on [ n ] and K m on [ m ] for n ≥ 2 and m ≥ 2 with (n , m) ≠ (2 , 2) , is equal to n + m + 2. As a corollary, we find that the NF -number of the complete bipartite graph K n , m on [ n + m ] is also equal to n + m + 2. [ABSTRACT FROM AUTHOR]

In this paper, we construct a class of association schemes by using pairs of subspaces of vector spaces and determine their full automorphism groups. [ABSTRACT FROM AUTHOR]

In this paper we introduce a particular semigroup transform A that fixes the invariants involved in Wilf's conjecture, except the embedding dimension. It also allows one to arrange the set of non-ordinary and non-irreducible numerical semigroups in a family of rooted trees. In addition, we study another transform, having similar features, that has been introduced by Bras-Amorós, and we make a comparison of them. In particular, we study the behavior of the embedding dimension under the action of such transforms, providing some consequences concerning Wilf's conjecture. [ABSTRACT FROM AUTHOR]

The Lie algebra s l 2 (C) may be regarded in a natural way as a subalgebra of the infinite-dimensional Virasoro Lie algebra, so it is natural to consider connections between the representation theory of the two algebras. In this paper, we explore the restriction to s l 2 (C) of certain induced modules for the Virasoro algebra. Specifically, we consider Virasoro modules induced from so-called polynomial subalgebras, and we show that the restriction of these modules results in twisted versions of familiar modules such as Verma modules and Whittaker modules. [ABSTRACT FROM AUTHOR]

In this paper we obtain some equivalent conditions for the core invertibility and EP-ness of 1 − p q , 1 − p q p , p − p q and p − q , where p , q are projections in different settings, such as ∗-rings, ∗-reducing rings and C ∗ -algebras. Moreover, several representations for the core inverses of product, difference and sum of two generalized projections are derived. In particular, a number of examples are given to illustrate our results. [ABSTRACT FROM AUTHOR]

An element x of a finite group G is said to be primary if the order of x is a prime power. We define csp2 (G) as follows: if | x G | is a prime power for every primary element x of G , where x G is the conjugacy class of x in G , then csp2 (G) = 0 ; if there exists a primary element x in G such that | x G | is divisible by at least two distinct primes, then csp2 (G) = max { | x G |   |   x ∈ G is primary , | x G | is divisible by at least two distinct primes }. In this paper we discuss the influence of the number csp2 (G) on the structure of G. [ABSTRACT FROM AUTHOR]

Let R be a ring with non-zero identity. The unit graph of R , denoted by G (R) , is an undirected graph with all the elements of R as vertices and where distinct vertices x , y are adjacent if and only if x + y is a unit of R. In this paper, we investigate the Wiener index and hyper-Wiener index of G (R) and explicitly determine their values. [ABSTRACT FROM AUTHOR]

This is the first part of a series of two articles. In this paper we enumerate and classify the left braces of size p 2 q , where p and q are distinct prime numbers, by the classification of regular subgroups of the holomorph of the abelian groups of the same order. We also provide the formulas that define the constructed braces. [ABSTRACT FROM AUTHOR]

Let be an abelian category and M   ∈ A. Then M is called a (D 4) -object if, whenever A and B are subobjects of M with M   =   A   ⊕   B and f : A → B is an epimorphism, Ker ⁡ f is a direct summand of A. In this paper we give several equivalent conditions of (D 4) -objects in an abelian category. Among other results, we prove that any object M in an abelian category is (D 4) if and only if for every subobject K of M such that K is the intersection K 1 ∩ K 2 of perspective direct summands K 1 and K 2 of M with M   =   K 1   +   K 2 , every morphismr φ :   M   →   M / K can be lifted to an endomorphism θ : M → M in E n d A (M). [ABSTRACT FROM AUTHOR]

In Enochs' relative homological dimension theory occur the (co)resolvent and (co)proper dimensions, which are defined by proper and coproper resolutions constructed by precovers and preenvelopes, respectively. Recently, some authors have been interested in relative homological dimensions defined by just exact sequences. In this paper, we contribute to the investigation of these relative homological dimensions. First we study the relation between these two kinds of relative homological dimensions and establish some transfer results under adjoint pairs. Then relative global dimensions are studied, which lead to nice characterizations of some properties of particular cases of self-orthogonal subcategories. At the end of this paper, relative derived functors are studied and generalizations of some known results of balance for relative homology are established. [ABSTRACT FROM AUTHOR]

In this paper we investigate a categorical aspect of n-trivial extension of a ring by a family of modules. Namely, we introduce the right (resp., left) n-trivial extension of a category by a family of endofunctors. Among other results, projective, injective and flat objects of this category are characterized, and two applications are presented at the end of this paper. We characterize when an n-trivial extension ring is k-perfect and establish a result on the self-injective dimension of an n-trivial extension ring. [ABSTRACT FROM AUTHOR]

Let p be an odd prime, and n , k be nonnegative integers. Let D n , k (1 , x) be the reversed Dickson polynomial of the (k + 1) -th kind. In this paper, by using Hermite's criterion, we study the permutational properties of the reversed Dickson polynomials D n , k (1 , x) over finite fields in the case of n = m p s with 0 < m < p − 1. In particular, we provide some precise characterizations for D n , k (1 , x) being permutation polynomials over finite fields with characteristic p when n = 2 p s , or n = 3 p s , or n = 4 p s. [ABSTRACT FROM AUTHOR]

Let A be a completely decomposable homogeneous torsion-free abelian group of rank n (n ≥ 2). Let Θ m (n) = A ⋊ ⟨ α ⟩ be the split extension of A by an automorphism α which is a cyclic permutation of the direct components twisted by a rational integer m. Then Θ m (n) is an infinite soluble group. In this paper, the residual finiteness of Θ m (n) is investigated. [ABSTRACT FROM AUTHOR]

A set { a 1 , a 2 , ... , a m } of positive integers is called a Diophantine m -tuple if a i a j + 1 is a perfect square for all 1 ≤ i < j ≤ m. Let { a , b , c } be the Diophantine triple with c > max { a , b }. In this paper, we find the condition for the reduction of third element c , and using this result, we prove the extendibility of Diophantine pair { F k − 1 F k + 1 , F k − 2 F k + 2 } , where F n is the n -th Fibonacci number. [ABSTRACT FROM AUTHOR]

Jordan D-bialgebras were introduced by Zhelyabin. In this paper, we use a new approach to study Jordan D-bialgebras by a new notion of the dual representation of the regular representation of a Jordan algebra. Motivated by the essential connection between Lie bialgebras and Manin triples, we give an explicit proof of the equivalence between Jordan D-bialgebras and a class of special Jordan–Manin triples called double constructions of pseudo-euclidean Jordan algebras. We also show that a Jordan D-bialgebra leads to the Jordan Yang–Baxter equation under the coboundary condition and an antisymmetric nondegenerate solution of the Jordan Yang–Baxter equation corresponds to an antisymmetric bilinear form, which we call a Jordan symplectic form on Jordan algebras. Furthermore, there exists a new algebra structure called pre-Jordan algebra on Jordan algebras with a Jordan symplectic form. [ABSTRACT FROM AUTHOR]

This paper introduces the notion of depth with respect to ideals for unbounded DG-modules, and gives a reduction formula and the local nature of this depth. As applications, we provide several bounds of the depth in special cases, and recover and generalize the known results about the depth of complexes. In addition, the width with respect to ideals for unbounded DG-modules is investigated and the depth and width formulas for DG-modules are generalized. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring and U (R) the multiplicative group of unit elements of R. In 2012, Khashyarmanesh et al. defined the generalized unit and unitary Cayley graph, Γ (R , G , S) , corresponding to a multiplicative subgroup G of U (R) and a nonempty subset S of G with S − 1 = { s − 1   |   s ∈ S } ⊆ S , as the graph with vertex set R and two distinct vertices x and y being adjacent if and only if there exists s ∈ S such that x + s y ∈ G. In this paper, we characterize all Artinian rings R for which Γ (R , U (R) , S) is projective. This leads us to determine all Artinian rings whose unit graphs, unitary Cayley graphs and co-maximal graphs are projective. In addition, we prove that for an Artinian ring R for which Γ (R , U (R) , S) has finite nonorientable genus, R must be a finite ring. Finally, it is proved that for a given positive integer k , the number of finite rings R for which Γ (R , U (R) , S) has nonorientable genus k is finite. [ABSTRACT FROM AUTHOR]

A Bihom-Lie algebra is a generalized Hom-Lie algebra endowed with two commuting multiplicative linear maps. In this paper, we study representations of Bihom-Lie algebras. In particular, derivations, central extensions, derivation extensions, the trivial representation and the adjoint representation of Bihom-Lie algebras are studied in detail. [ABSTRACT FROM AUTHOR]