Search

In this paper, we study the Cartesian product of signed graphs as defined by Germina, Hameed and Zaslavsky (2011). Here we focus on its algebraic properties and look at the chromatic number of some Cartesian products. One of our main results is the unicity of the prime factor decomposition of signed graphs. This leads us to present an algorithm to compute this decomposition in linear time based on a decomposition algorithm for oriented graphs by Imrich and Peterin (2018). We also study the chromatic number of a signed graph, that is the minimum order of a signed graph to which the input signed graph admits a homomorphism, of graphs with underlying graph of the form P n □ P m , of Cartesian products of signed paths, of Cartesian products of signed complete graphs and of Cartesian products of signed cycles.

A graph database is a digraph whose arcs are labeled with symbols from a fixed alphabet. A regular graph pattern (RGP) is a digraph whose edges are labeled with regular expressions over the alphabet. RGPs model navigational queries for graph databases called conjunctive regular path queries (CRPQs). A match of a CRPQ in the database is witnessed by a special navigational homomorphism of the corresponding RGP to the database. We study the complexity of deciding the existence of a homomorphism between two RGPs. Such homomorphisms model a strong type of containment between the two corresponding CRPQs. We show that this problem can be solved by an EXPTIME algorithm (while general query containmement in this context is EXPSPACE-complete). We also study the problem for restricted RGPs over a unary alphabet, that arise from some applications like XPath or SPARQL. For this case, homomorphism-based CRPQ containment is in NP. We prove that certain interesting cases are in fact polynomial-time solvable., 15 pages. Short version appeared in the proceedings of the 15th Conference on Computability in Europe (CIE 2019)

This paper introduces the concept of homomorphism on fuzzy hyperlattice ordered group ( FHLOG ) . It studies how the binary and the fuzzy hyperoperations of a FHLOG can be transformed into the binary and the fuzzy hyperoperations of another FHLOG . The notion of fuzzy hypercongruence relation on FHLOG is also defined. The paper also establishes the redox reaction of copper, gold and americium forms three FHLOG s. Besides, homomorphism and composition function of FHLOG s using the redox reactions are developed. Therefore, the paper develops a relation among three different metal’s redox reactions in which the binary and the fuzzy hyperoperations, are preserved.

We study Gorenstein flat objects in the category Rep ( Q , R ) of representations of a left rooted quiver Q with values in Mod ( R ) , the category of all left R-modules, where R is an arbitrary associative ring. We show that a representation X in Rep ( Q , R ) is Gorenstein flat if and only if for each vertex i the canonical homomorphism φ i X : ⊕ a : j → i X ( j ) → X ( i ) is injective, and the left R-modules X ( i ) and Coker φ i X are Gorenstein flat. As an application, we obtain a Gorenstein flat model structure on Rep ( Q , R ) in which we give explicit descriptions of the subcategories of trivial, cofibrant and fibrant objects.

The study of operator algebras on Hilbert spaces, and C ∗ -algebras in particular, is one of the most active areas within Functional Analysis. A natural generalization of these is to replace Hilbert spaces (which are L 2 -spaces) with L p -spaces, for p ∈ [ 1 , ∞ ) . The study of such algebras of operators is notoriously more challenging, due to the very complicated geometry of L p -spaces by comparison with Hilbert spaces. We give a modern overview of a research area whose beginnings can be traced back to the 50’s, and that has seen renewed attention in the last decade through the infusion of new techniques. The combination of these new ideas with old tools was the key to answer some long standing questions. Among others, we provide a description of all unital contractive homomorphisms between algebras of p -pseudofunctions of groups.

The main purpose of this article is to initiate a systematic study of Semihypergroups, first introduced by C. Dunkl [4], I. Jewett [13] and R. Spector [20] independently around 1972. We introduce and study several natural algebraic and analytic structures on semihypergroups, which are well-known in the case of topological groups and semigroups. In particular, we first study almost periodic and weakly almost periodic function spaces (basic properties, their relation to the compactness of the underlying space, introversion and Arens product on their duals among others). We then introduce homomorphisms and ideals, and thereby examine their behaviour (basic properties, structure of the kernel and relation of amenability to minimal ideals) in order to gain insight into the structure of a Semihypergroup itself. In the process, we further investigate where and why this theory deviates from the classical theory of semigroups., 26 Pages, comments are welcome

We first introduce the class of bipartite absolute retracts with respect to tree obstructions with at most k leaves. Then, using the theory of homomorphism duality, we show that this class of absolute retracts coincides exactly with the bipartite graphs which admit a $$(k+1)$$ -ary near-unanimity polymorphism. This result mirrors the case for reflexive graphs and generalizes a known result for bipartite graphs which admit a 3-ary near-unanimity polymorphism.

Every Krull monoid has a transfer homomorphism onto a monoid of zero-sum sequences over a subset of its class group. This transfer homomorphism is a crucial tool for studying the arithmetic of Krull monoids. In the present paper, we strengthen and refine this tool for Krull monoids with finitely generated class group.

A recent result of G. Cz\'edli relates the ordered set of principal congruences of a bounded lattice $L$ with the ordered set of principal congruences of a~bounded sublattice $K$ of $L$. In this note, I sketch a new proof.

We propose an algorithm for linearizing systems of partial differential equations at constant solutions. The algorithm is based on an isomorphism constructed between the ring of linearized functions and the ring of special matrices, which makes it possible to simplify calculations in the process of linearization. The algorithm is illustrated by applying it to the quasigasdynamic system.

Denote a finite dimensional Hopf C*-algebra by H, and a Hopf *-subalgebra of H by H1. In this paper, we study the construction of the field algebra in Hopf spin models determined by H1 together with its symmetry. More precisely, we consider the quantum double D(H, H1) as the bicrossed product of the opposite dual $$\widehat{{H^{op}}}$$ of H and H1 with respect to the coadjoint representation, the latter acting on the former and vice versa, and under the non-trivial commutation relations between H1 and Ĥ we define the observable algebra $${{\cal A}_{{H_1}}}$$ . Then using a comodule action of D(H, H1) on $${{\cal A}_{{H_1}}}$$ , we obtain the field algebra $${{\cal F}_{{H_1}}}$$ , which is the crossed product $${{\cal A}_{{H_1}}} \rtimes D\widehat{\left({H,{H_1}} \right)}$$ , and show that the observable algebra $${{\cal A}_{{H_1}}}$$ is exactly a D(H, H1)-invariant subalgebra of $${{\cal F}_{{H_1}}}$$ . Furthermore, we prove that there exists a duality between D(H, H1) and $${{\cal A}_{{H_1}}}$$ , implemented by a *-homomorphism of D(H, H1).

We suggest an algorithm for constructing all the isolated primary components of a given polynomial ideal. At the output, they are determined by systems of generators up to embedded components, and also as kernels of some homomorphisms. The complexity of this algorithm is subexponential in the size of the input data.

Sets of solutions to finite systems of equations in a free group, are equivalent to sets of homomorphisms from a fixed f.p. group into a free group. The latter can be encoded in a diagram, the construction of which is valid also for f.g. groups. The diagram is known to be canonical for a fixed f.g. group with a fixed generating set. In this paper we prove that the construction depends on the chosen generating set of the given f.g. group.

This paper shall be concerned with the notion of von-Neumann regular acts. First, the notions of semiretraction homomorphisms and fully semiretractable (FSR) S-acts are introduced and some properties of them are investigated. Finally, using semiretraction notion, we investigate locally projective and von-Neumann regular S-acts.

A hom-Lie algebroid is a vector bundle together with a Lie algebroid-like structure which is twisted by a homomorphism and a VB-hom algebroid is essentially defined as a hom-Lie algebroid object in the category of vector bundles. In this paper, we show that, there exists a correspondence between the VB-hom algebroids and two term representations up to homotopy of hom-Lie algebroid.

Let $$(S,\le )$$ be a strictly totally ordered monoid and R an $$(S,\omega )$$ -weakly rigid ring, where $$\omega :S\rightarrow End(R)$$ is a monoid homomorphism. In this paper, we study the weakly p.q.-Bear property of the skew generalized power series ring $$R[[S,\omega ]]$$ . As a consequence, the weakly p.q.-Baer property of the skew power series ring $$R[[x;\alpha ]]$$ and the skew Laurent power series ring $$R[[x,x^{-1};\alpha ]]$$ are determined, where $$\alpha$$ is a ring endomorphism of R.

UDC 512.5Smith in paper [Mapping between module lattices, Int. Electron. J. Algebra, 15, 173–195 (2014)] introduced maps between the lattice of ideals of a commutative ring and the lattice of submodules of an -module i.e., and mappings.The definitions of the maps were motivated by the definition of multiplication modules.Moreover, some sufficient conditions for the maps to be a lattice homomorphisms are studied.In this work we define a class of -modules and observe the properties of the class. We give a sufficient conditions for the module and the ring such that the class is a hereditary pretorsion class.

In this paper we study graded Bourbaki ideals. It is a well-known fact that for torsionfree modules over Noetherian normal domains, Bourbaki sequences exist. We give criteria in terms of certain attached matrices for a homomorphism of modules to induce a Bourbaki sequence. Special attention is given to graded Bourbaki sequences. In the second part of the paper, we apply these results to the Koszul cycles of the residue class field and determine particular Bourbaki ideals explicitly. We also obtain in a special case the relationship between the structure of the Rees algebra of a Koszul cycle and the Rees algebra of its Bourbaki ideal., Comment: 29 pages

In this paper, at first we study strong Sheffer stroke NMV-algebra. For getting more results and some classification, the notions of filters and subalgebras are introduced and studied. Finally, by a congruence relation, we construct a quotient strong Sheffer stroke NMV-algebra and isomorphism theorems are proved., Ege University Scientific Research Projects Directorate [20772], We wish to thank the reviewers for excellent suggestions that have been incorporated into the paper. This study is supported by Ege University Scientific Research Projects Directorate with the project number 20772.

The paper is devoted to the normal extensions of discrete semigroups and $ * $ -homomorphisms of semigroup $ C^{*} $ -algebras. We study the normal extensions of abelian semigroups by arbitrary groups. Considering numerical semigroups, we prove that they are normal extensions of the semigroup of nonnegative integers by finite cyclic groups. On the other hand, we prove that if a semigroup is a normal extension of the semigroup of nonnegative integers by a finite cyclic group generated by a single element then this semigroup is isomorphic to a numerical semigroup. As regard a normal extension with a generating set, we consider two reduced semigroup $ C^{*} $ -algebras defined by this extension. We show that there exists an embedding of the semigroup $ C^{*} $ -algebras which is generated by an injective homomorphism of the semigroups and the natural isometric representations of these semigroups.

Recently, Wickstead investigated the long-standing problem of adding an identity to a non-unital Banach lattice algebra. In this regard, he proved that, in the category whose objects are unital Banach lattice algebras and morphisms are identity preserving algebra and lattice homomorphisms, there is no reflection for $$c_{0}$$ in which $$c_{0}$$ can be embedded as an algebra and order ideal. The main purpose of this note is to describe an alternative category in which a suitable reflection of $$c_{0}$$ can be located, producing a satisfactory lattice unitization of $$c_{0}$$ with $$c_{0}$$ as an algebra and order ideal.

In this paper, we continue to study and investigate the homological objects related to s-pure and neat exact sequences of modules and module homomorphisms. A right module A is called max-flat if $${\text {Tor}}_{1}^{R}(A, R/I)= 0$$ for any maximal left ideal I of R. A right module B is said to be max-cotorsion if $${\text {Ext}}^{1}_{R}(A,B)=0$$ for any max-flat right module A. We characterize some classes of rings such as perfect rings, max-injective rings, SF rings and max-hereditary rings by max-flat and max-cotorsion modules. We prove that every right module has a max-flat cover and max-cotorsion envelope. We show that a left perfect right max-injective ring R is QF if and only if maximal right ideals of R are finitely generated. The max-flat dimensions of modules and rings are studied in terms of right derived functors of $$-\otimes -$$ . Finally, we study the modules that are injective and flat relative to s-pure exact sequences.

We show that for every sofic chunk $E$ there is a bijective homomorphism $f: E_c \rightarrow E$, where $E_c$ is a chunk of the group of computable permutations of $\mathbb{N}$ so that the approximating morphisms of $E$ can be viewed as restrictions of permutations of $E_c$ to finite subsets of $\mathbb{N}$. Using this we study some relevant effectivity conditions associated with sofic chunks and their profiles., Comment: This is a revised version of the paper, 21 pages

Let $M = \mathbb{B}^2 / \Gamma$ be a smooth ball quotient of finite volume with first betti number $b_1(M)$ and let $\mathcal{E}(M) \ge 0$ be the number of cusps (i.e., topological ends) of $M$. We study the growth rates that are possible in towers of finite-sheeted coverings of $M$. In particular, $b_1$ and $\mathcal{E}$ have little to do with one another, in contrast with the well-understood cases of hyperbolic $2$- and $3$-manifolds. We also discuss growth of $b_1$ for congruence arithmetic lattices acting on $\mathbb{B}^2$ and $\mathbb{B}^3$. Along the way, we provide an explicit example of a lattice in $\mathrm{PU}(2, 1)$ admitting a homomorphism onto $\mathbb{Z}$ with finitely generated kernel. Moreover, we show that any cocompact arithmetic lattice $\Gamma \subset \mathrm{PU}(n, 1)$ of simplest type contains a finite index subgroup with this property.

In this paper, we generalize the concept of homomorphism from BE-semigroups to generalized BE-semigroups. In order to show the existence, we construct some examples. Furthermore, we characterize generalized BE-semigroups by using homomorphism. In particular, we show that through every homomorphism, we may get factor generalized BE-semigroup.

If H is a subgroup of an Abelian p-group G, we say G is H-fully transitive if using the height valuation from G, for every x ∈ H , every valuated (i.e., non-height decreasing) homomorphism 〈 x 〉 → G extends to a valuated homomorphism H → G . This notion is a generalization of the classical definition of fully transitive groups due to Kaplansky. A number of interesting properties of this idea are established. For example, a complete characterization of those valuated groups H that are universally fully transitive in the sense that every group G that contains H as such an embedded subgroup is necessarily H-fully transitive. Particular attention is given to the case where H is isotype in G, so the induced valuation agrees with the height valuation on H. Our results concerning the full transitivity of Abelian p-groups somewhat enlarge those obtained by Griffith (1968) [9] and Goldsmith and Strungmann (2007) [8] . Our other results concerning the valuated Abelian p-groups somewhat refine those established by Richman and Walker (1979) [17] .

Let $${\Lambda _{(0,0)}} = \left( {\matrix{A \;\;\;\;\;\;\;\; {_A{N_B}} \cr {_B{M_A}}\;\;\;\; B \cr } } \right)$$ be a Morita ring, where the bimodule homomorphisms ϕ and ψ are zero. We study the finite presentedness, locally coherence, pure projectivity, pure injectivity, and FP-injectivity of modules over Λ(0,0). Some applications are then given.

We apply the recently introduced framework of admissible homomorphisms in the form of a convolution algebra of $$\mathbb{C}^2$$ -valued admissible homomorphisms to handle two-dimensional uni-directional homogeneous stochastic kernels. The algebra product is a non-commutative extension of the Feller convolution needed for an adequate operator representation of such kernels: a pair of homogeneous transition functions uni-directionally intertwined by the extended Chapman–Kolmogorov equation is a convolution empathy; the associated Fokker–Planck equations are re-written as an implicit Cauchy equation expressed in terms of admissible homomorphisms. The conditions of solvability of such implicit evolution equations follow from the consideration of generators of a convolution empathy.

The famous unit conjecture for group algebras states that every unit is trivial. The validity of this conjecture is not known for the sightly simple example of fours group $$\Gamma =\langle x, y\; \vert \; (x^2)^y=x^{-2}, (y^2)^x=y^{-2}\rangle $$ which it is “the simplest” example of a torsion-free non unique-product supersoluble group. In this article for $$n\in \mathbb {N}$$ , we set $$H_n=\langle x^{2^n}, y^{2^n}, (xy)^{2^n}\rangle \leqslant \Gamma $$ and we consider $$G_n=\Gamma /H_n$$ . We will show that there is a large subset $$\mathcal {N}_n$$ of $$\mathbb {C}[G_n]$$ which its elements are non-unit, so all elements of the set $$\mathcal {N}=\bigcup _{n\in \mathbb {N}}\varphi _n^{-1}(\mathcal {N}_n)$$ are non-unit in $$\mathbb {C}[\Gamma ]$$ , where $$\varphi _n: \mathbb {C}[\Gamma ]\rightarrow \mathbb {C}[G_n]$$ is the induced group ring homomorphism by the quotient map $$\varphi _n: \Gamma \rightarrow G_n$$ .

The purpose of this work is to use the concepts of reachability and coreachability maps to provide a solution of a well-known characterization for [Formula: see text]-valued multiset regular languages. In between, we associate two deterministic [Formula: see text]-valued multiset automata with a given deterministic [Formula: see text]-valued multiset automaton (DLMA) and show that the reachability and coreachability maps of the given DLMA turn out to be morphisms in the category of deterministic [Formula: see text]-valued multiset automata.

Consider a compact symplectic sub-orbifold groupoid $\sf S$ of a compact symplectic orbifold groupoid $(\mathsf X,\omega)$. Let $\mathsf X_{\mathfrak a}$ be the weight-$\mathfrak a$ blowup of $\sf X$ along $\sf S$, and $\mathsf D_{\mathfrak a}=\mathsf{PN}_{\mathfrak a}$ be the exceptional divisor, where $\sf N$ is the normal bundle of $\sf S$ in $\sf X$. In this paper we show that the absolute orbifold Gromov--Witten theory of $\mathsf X_{\mathfrak a}$ can be effectively and uniquely reconstructed from the absolute orbifold Gromov--Witten theories of $\sf X$, $\sf S$ and $\mathsf D_{\mathfrak a}$, the natural restriction homomorphism $H^*_{\text{CR}}({\sf X})\rightarrow H^*_{\text{CR}}({\sf S})$ and the first Chern class of the tautological line bundle over $\mathsf D_{\mathfrak a}$. To achieve this we first prove similar results for the relative orbifold Gromov--Witten theories of $(\mathsf X_{\mathfrak a}|\mathsf D_{\mathfrak a})$ and $(\mathsf N_{\mathfrak a}|\mathsf D_{\mathfrak a})$. As applications of these results, we prove an orbifold version of a conjecture of Maulik--Pandharipande on the Gromov--Witten theory of blowups along complete intersections, a conjecture on the Gromov--Witten theory of root constructions and a conjecture on Leray--Hirsch result for orbifold Gromov--Witten theory of Tseng--You., Comment: 44pages; some typos fixed; comments are welcome; accepted for publication in SCIENCE CHINA Mathematics

A module M is called a D4-module if, whenever A and B are submodules of M with M = A ⊕ B and f : A → B is a homomorphism with Imf a direct summand of B, then Kerf is a direct summand of A. The class of D4-modules contains the class of D3-modules, and hence the class of semi-projective modules, and so the class of Rickart modules. In this paper we prove that, over a commutative Dedekind domain R, for an R-module M which is a direct sum of cyclic submodules, M is direct projective (equivalently, it is semi-projective) iff M is D3 iff M is D4. Also we prove that, over a prime PI-ring, for a divisible R-module X, X is direct projective (equivalently, it is Rickart) iff X ⊕ X is D4. We determine some D3-modules and D4-modules over a discrete valuation ring, as well. We give some relevant examples. We also provide several examples on D3-modules and D4-modules via quivers.

We give asymptotics for the cumulative distribution function (CDF) for degrees of large dense random graphs sampled from a graphon. The proof is based on precise asymptotics for binomial random variables. Replacing the indicator function in the empirical CDF by a smoother function, we get general asymptotic results for functionals of homomorphism densities for partially labeled graphs with smoother functions. This general setting allows to recover recent results on asymptotics for homomorphism densities of sampled graphon., 49 pages, 3 figures

In this paper we recast the proof of Rossman's equirank homomorphism preservation theorem using comonadic formulations of bounded quantifier rank and variable count (and dually tree width and tree-depth), and work towards generalisation of it that simultaneously preserves quantifier rank and variable count. Along the way, we give an exposition of the required comonads, showing how their properties arise.

In 2006, Schuster introduced the radial Blaschke-Minkowski homomorphisms. In this article, associating with the star duality of star bodies and dual quermassintegrals, we establish Brunn-Minkowski inequalities and monotonic inequality for the radial Blaschke-Minkowski homomorphisms. In addition, we consider its Shephard-type problems and give a positive form and a negative answer, respectively.

First we give a definition of a coverage on a inverse semigroup that is weaker than the one gave by a Lawson and Lenz and that generalizes the definition of a coverage on a semilattice given by Johnstone. Given such a coverage, we prove that there exists a pseudogroup that is universal in the sense that it transforms cover-to-join idempotent-pure maps into idempotent-pure pseudogroup homomorphisms. Then, we show how to go from a nucleus on a pseudogroup to a topological groupoid embedding of the corresponding groupoids. Finally, we apply the results found to study Exel's notions of tight filters and tight groupoids.

We study the vector-valued spectrum $${\mathcal {M}}_\infty (B_{c_0},B_{c_0})$$ , that is, the set of nonzero algebra homomorphisms from $$\mathcal {H}^\infty (B_{c_0})$$ to $$\mathcal {H}^\infty (B_{c_0})$$ which is naturally projected onto the closed unit ball of $$\mathcal {H}^\infty (B_{c_0}, \ell _\infty )$$ , likewise the scalar-valued spectrum $$\mathcal {M}_\infty (B_{c_0})$$ which is projected onto $$\overline{B}_{\ell _\infty }$$ . Our itinerary begins in the scalar-valued spectrum $${\mathcal {M}}_\infty (B_{c_0})$$ : by expanding a result by Cole et al. (Michigan Math J 39(3):551–569, 1992), we prove that in each fiber, there are $$2^c$$ disjoint analytic Gleason isometric copies of $$B_{\ell _\infty }$$ . For the vector-valued case, building on the previous result we obtain $$2^c$$ disjoint analytic Gleason isometric copies of $$B_{{\mathcal {H}}^\infty (B_{c_0},\ell _\infty )}$$ in each fiber. We also take a look at the relationship between fibers and Gleason parts for both vector-valued spectra $${\mathcal {M}}_{u,\infty }(B_{c_0},B_{c_0})$$ and $${\mathcal {M}}_\infty (B_{c_0},B_{c_0})$$ .

Let $\Sigma$ be a surface with either boundary or marked points, equipped with an arbitrary framing. In this note we determine the action of the associated "framed mapping class group" on the homology of $\Sigma$ relative to its boundary (respectively marked points), describing the image as the kernel of a certain crossed homomorphism related to classical spin structures. Applying recent work of the authors, we use this to describe the monodromy action of the orbifold fundamental group of a stratum of abelian differentials on the relative periods., Comment: 18 pages. Final version, as appeared in B. Lond. Math. Soc

We would like to build Abelian groups (or R-modules) which on the one hand are quite free, say ℵω+1-free, and on the other hand are complicated in a suitable sense. We choose as our test problem one having no nontrivial homomorphism to ℤ (known classically for ℵ1-free, recently for ℵn-free). We succeed to prove the existence of even $${\aleph _{{\omega _1} \cdot n}}$$ -free ones. This requires building n-dimensional black boxes, which are quite free. This combinatorics is of self interest and we believe will be useful also for other purposes. On the other hand, modulo suitable large cardinals, we prove that it is consistent that every $${\aleph _{{\omega _1} \cdot \omega }}$$ -free Abelian group has non-trivial homomorphisms to ℤ.

We study homomorphism problems of signed graphs. A signed graph is an undirected graph where each edge is given a sign, positive or negative. An important concept for signed graphs is the operation of switching at a vertex, which is to change the sign of each incident edge. A homomorphism of a graph is a vertex-mapping that preserves the adjacencies; in the case of signed graphs, we also preserve the edge-signs. Special homomorphisms of signed graphs, called s-homomorphisms, have been studied. In an s-homomorphism, we allow, before the mapping, to perform any number of switchings on the source signed graph. This concept has been extensively studied, and a full complexity classification (polynomial or NP-complete) for s-homomorphism to a fixed target signed graph has recently been obtained. Such a dichotomy is not known when we restrict the input graph to be planar (not even for non-signed graph homomorphisms). We show that deciding whether a (non-signed) planar graph admits a homomorphism to the square $C_t^2$ of a cycle with $t\ge 6$, or to the circular clique $K_{4t/(2t-1)}$ with $t\ge2$, are NP-complete problems. We use these results to show that deciding whether a planar signed graph admits an s-homomorphism to an unbalanced even cycle is NP-complete. (A cycle is unbalanced if it has an odd number of negative edges). We deduce a complete complexity dichotomy for the planar s-homomorphism problem with any signed cycle as a target. We also study further restrictions involving the maximum degree and the girth of the input signed graph. We prove that planar s-homomorphism problems to signed cycles remain NP-complete even for inputs of maximum degree~$3$ (except for the case of unbalanced $4$-cycles, for which we show this for maximum degree~$4$). We also show that for a given integer $g$, the problem for signed bipartite planar inputs of girth $g$ is either trivial or NP-complete., Comment: 17 pages, 10 figures

We generalize chain enumeration in graded partially ordered sets by relaxing the graded, poset and Eulerian requirements. The resulting balanced digraphs, which include the classical Eulerian posets having an R-labeling, imply the existence of the (non-homogeneous) $$\mathbf{c}\mathbf{d}$$ -index, a key invariant for studying inequalities for the flag vector of polytopes. Mirroring Alexander duality for Eulerian posets, we show an analogue of Alexander duality for bounded balanced digraphs. For Bruhat graphs of Coxeter groups, an important family of balanced graphs, our theory gives elementary proofs of the existence of the complete $$\mathbf{c}\mathbf{d}$$ -index and its properties. We also introduce the rising and falling quasisymmetric functions of a labeled acyclic digraph and show they are Hopf algebra homomorphisms mapping balanced digraphs to the Stembridge peak algebra. We conjecture non-negativity of the $$\mathbf{c}\mathbf{d}$$ -index for acyclic digraphs having a balanced linear edge labeling.

Let $${\mathcal {R}}$$ be a semiprime ring with center $$Z({\mathcal {R}})$$ and with extended centroid C and let $$\sigma : {\mathcal {R}} \rightarrow {\mathcal {R}}$$ be an automorphism. Assume that $$\tau : {\mathcal {R}} \rightarrow {\mathcal {R}} $$ is an anti-homomorphism, such that the image of $$\tau $$ has small centralizer. It is proved that the following are equivalent: (1) $$x^{\sigma }x^{\tau } = x^{\tau }x^{\sigma }$$ for all $$x\in {\mathcal {R}};$$ (2) $$x^{\sigma } + x^{\tau }\in Z({\mathcal {R}})$$ for all $$x\in {\mathcal {R}};$$ (3) $$x^{\sigma }x^{\tau }\in Z({\mathcal {R}})$$ for all $$x\in {\mathcal {R}}.$$ In this case, there exists an idempotent $$e \in C$$ , such that $$(1-e){\mathcal {R}}$$ is a commutative ring and the semiprime ring $$e{\mathcal {R}}$$ is equipped with an involution $$\widetilde{\tau }$$ , which is induced canonically by $$\tau $$ . Note that one can easily obtained the main result in Lee (Commun Algebra 46(3):1060–1065, 2018) when $$\sigma =id_{{\mathcal {R}}}.$$

The aim of this article is to determine entirely the Jordan automorphisms of generalized matrix rings of Morita contexts. Necessary and sufficient conditions are obtained for an $$\mathcal {R}$$ -linear map on a general Morita context to be a Jordan homomorphism. Moreover, some conditions are studied, under which, any Jordan automorphism of a general Morita context is either an automorphism or an anti-automorphism.

By a theorem of Banagl-Chriestenson, intersection spaces of depth one pseudomanifolds exhibit generalized Poincar\'{e} duality of Betti numbers, provided that certain characteristic classes of the link bundles vanish. In this paper, we show that the middle-perversity intersection space of a depth one Witt space can be completed to a rational Poincar\'{e} duality space by means of a single cell attachment, provided that a certain rational Hurewicz homomorphism associated to the link bundles is surjective. Our approach continues previous work of Klimczak covering the case of isolated singularities with simply connected links. For every singular stratum, we show that our condition on the rational Hurewicz homomorphism implies that the Banagl-Chriestenson characteristic classes of the link bundle vanish. Moreover, using Sullivan minimal models, we show that the converse implication holds at least in the case that twice the dimension of the singular stratum is bounded by the dimension of the link. As an application, we compare the signature of our rational Poincar\'{e} duality space to the Goresky-MacPherson intersection homology signature of the given Witt space. We discuss our results for a class of Witt spaces having circles as their singular strata., Comment: 32 pages

We study and classify the 3-dimensional Hom-Lie algebras over $\mathbb{C}$. We provide first a complete set of representatives for the isomorphism classes of skew-symmetric bilinear products defined on a 3-dimensional complex vector space $\mathfrak{g}$. The well known Lie brackets for the 3-dimensional Lie algebras are included into appropriate isomorphism classes of such products representatives. For each product representative, we provide a complete set of canonical forms for the linear maps $\mathfrak{g} \to \mathfrak{g}$ that turn $g$ into a Hom-Lie algebra, thus characterizing the corresponding isomorphism classes. As by-products, Hom-Lie algebras for which the linear maps $\mathfrak{g} \to \mathfrak{g}$ are not homomorphisms for their products, are exhibited. Examples also arise of non-isomorphic families of HomLie algebras which share, however, a fixed Lie-algebra product on $\mathfrak{g}$. In particular, this is the case for the complex simple Lie algebra $\mathfrak{sl}_2(\mathbb{C})$. Similarly, there are isomorphism classes for which their skew-symmetric bilinear products can never be Lie algebra brackets on $\mathfrak{g}$.

A meet in a frame is exact if it join-distributes with every element, it is strongly exact if it is preserved by every frame homomorphism. Hence, finite meets are (strongly) exact which leads to the concept of an exact resp. strongly exact filter, a filter closed under exact resp. strongly exact meets. It is known that the exact filters constitute a frame $${\mathrm{Filt}}_{{\textsf {E}}}(L)$$ somewhat surprisingly isomorphic to the frame of joins of closed sublocales. In this paper we present a characteristic of the coframe of meets of open sublocales as the dual to the frame of strongly exact filters $${\mathrm{Filt}}_{{\textsf {sE}}}(L)$$ .

Recall that a left module A (resp. right module B) is said to be singly injective (resp. singly flat) if $$Ext^{1}_{R}(F/K, A) = 0$$ (resp. $${\text {Tor}}_{1}^{R}(B, F/K)= 0$$ ) for any cyclic submodule K of any finitely generated free left R-module F. In this paper, we continue to study and investigate the homological objects related to singly flat and singly injective modules and module homomorphisms. Along the way, the right orthogonal class of singly flat right modules and the left orthogonal class of singly injective left modules are introduced and studied. These concepts are used to extend the some known results and to characterize pseudo-coherent rings and left singly injective rings. In terms of some derived functors, some homological dimensions are investigated. As applications, some new characterizations of von Neumann regular rings and left PP rings are given. Finally, we study the singly flatness and singly injectivity of homomorphism modules over a commutative ring.

In this article, some properties of the Schur multiplier and residual nilpotency of Lie rings are studied. In the case of nilpotent Lie rings and given special homomorphism between rings, some isom...

In this paper, we define the $K$-theoretic Hall algebra for dimension $0$ coherent sheaves on a smooth projective surface, prove that the algebra is associative, and construct a homomorphism to a shuffle algebra introduced by Negut [ 10].

Inspired by some recent development on the theory about projection valued dilations for operator valued measures or more generally bounded homomorphism dilations for bounded linear maps on Banach algebras, the authors explore a pure algebraic version of the dilation theory for linear systems acting on unital algebras and vector spaces. By introducing two natural dilation structures, namely the canonical and the universal dilation systems, they prove that every linearly minimal dilation is equivalent to a reduced homomorphism dilation of the universal dilation, and all the linearly minimal homomorphism dilations can be classified by the associated reduced subspaces contained in the kernel of synthesis operator for the universal dilation.