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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)

For a fixed digraph G, the Constraint Satisfaction Problem with the template G, or CSP(G) for short, is the problem of deciding whether a given input digraph H admits a homomorphism to G. The dichotomy conjecture of Feder and Vardi states that CSP(G), for any choice of G, is solvable in polynomial time or NP-complete. This paper confirms the conjecture for a class of oriented trees called special triads. As a corollary we get the smallest known example of an oriented tree (with 33 vertices) defining an NP-complete CSP(G).

У статті розглядаються розширені і стандартні описи гомоморфізмів груп E (n,R) ⊆G ⊆GL(n,R), n≥2 над асоціативними кільцями R з 1. Показано, що гомоморфізми з умовою (*) групи E (n,R) < G ⊆ GL(n,R), n≥4 над асоціативними кільцями R з 1 мають розширено стандартний опис, а при деяких обмеженнях стандартний опис на групах G і E(n,R). В роботі також описуються гомоморфізми з умовою (*) групи (n,R) ⊆ G ⊆ GL(n,R), n≥4, що відображають її у групу GL(m,K), m≥2, які є мономорфізмами (зокрема такими є ізоморфізми) або E (n,K) ⊆ΛE (n,R) над асоціативними кільцями R і K з 1. Показано, що такі гомоморфізми допускають стандартний опис на групі E (n,R).

Let M_n (R_1 [[S_1,≤_1,ω_1]]) and M_n (R_2 [[S_2,≤_2,ω_2]]) be a matrix rings over skew generalized power series rings, where R_1,R_2 are commutative rings with an identity element, (S_1,≤_1 ),(S_2,≤_2 ) are strictly ordered monoids, ω_1:S_1→End(R_1 ),〖 ω〗_2:S_2→End(R_2 ) are monoid homomorphisms. In this research, a mapping τ from M_n (R_1 [[S_1,≤_1,ω_1]]) to M_n (R_2 [[S_2,≤_2,ω_2]]) is defined by using a strictly ordered monoid homomorphism δ:(S_1,≤_1 )→(S_2,≤_2 ), and ring homomorphisms μ:R_1→R_2 and σ:R_1 [[S_1,≤_1,ω_1]]→R_2 [[S_2,≤_2,ω_2]]. Furthermore, it is proved that τ is a ring homomorphism, and also the sufficient conditions for τ to be a monomorphism, epimorphism, and isomorphism are given.

Let C be an Abelian group. A class X of Abelian groups is called a CE• H-class if for any groups A, B ∈ X, it follows from the existence of isomorphisms E• (A) ≅ E• (B) and Hom(C,A) ≅ Hom(C,B) that there is an isomorphism A ≅ B. In this paper, conditions are studied under which the class $$ {\mathfrak{I}}_{\mathrm{cd}}^{\mathrm{ad}} $$ of completely decomposable almost divisible Abelian groups and class $$ {\mathfrak{I}}_{\mathrm{cd}}^{\ast } $$ of completely decomposable torsion-free Abelian groups A where Ω(A) contains only incomparable types are CE• H-classes, where C is a completely decomposable torsion-free Abelian group.

A multiplication on an Abelian group G is a homomorphism μ: G ⊗ G → G. An Abelian group G with a multiplication on it is called a ring on the group G. R. A. Beaumont and D. A. Lawver have formulated the problem of studying semisimple groups. An Abelian group is said to be semisimple if there exists a semisimple associative ring on it. Semisimple groups are described in the class of vector Abelian nonmeasurable groups. It is also shown that if a set I is nonmeasurable, $$ G=\coprod_{i\in I}{A}_i $$ is a reduced vector Abelian group, and μ is a multiplication on G, then μ is determined by its restriction on the sum $$ {\displaystyle \begin{array}{c}\oplus \\ {}i\in I\end{array}}{A}_i $$ this statement is incorrect if the set I is measurable or the group G is not reduced.

The Mordell-Weil groups $E(\mathbb{Q})$ of elliptic curves influence the structures of their quadratic twists $E_{-D}(\mathbb{Q})$ and the ideal class groups $\mathrm{CL}(-D)$ of imaginary quadratic fields. For appropriate $(u,v) \in \mathbb{Z}^2$, we define a family of homomorphisms $\Phi_{u,v}: E(\mathbb{Q}) \rightarrow \mathrm{CL}(-D)$ for particular negative fundamental discriminants $-D:=-D_E(u,v)$, which we use to simultaneously address questions related to lower bounds for class numbers, the structures of class groups, and ranks of quadratic twists. Specifically, given an elliptic curve $E$ of rank $r$, let $\Psi_E$ be the set of suitable fundamental discriminants $-D 0$, we show that as $X \to \infty$, we have $$\#\, \left\{-X < -D < 0: -D \in \Psi_E \right \} \, \gg_{\varepsilon} X^{\frac{1}{2}-\varepsilon}.$$ In particular, if $\ell \in \{3,5,7\}$ and $\ell \mid |E_{\mathrm{tor}}(\mathbb{Q})|$, then the number of such discriminants $-D$ for which $\ell \mid h(-D)$ is $\gg_{\varepsilon} X^{\frac{1}{2}-\varepsilon}.$ Moreover, assuming the Parity Conjecture, our results hold with the additional condition that the quadratic twist $E_{-D}$ has rank at least 2.

Aim of this study is to investigate anti-homomorphic images and pre-images of semiprime and primary ideals in interval valued fuzzy Near-rings. Further some results on f-invariant interval valued fuzzy ideal, f-invariant strongly primary interval valued fuzzy ideal and f-invariant semiprime interval valued fuzzy ideals of Near-rings are discussed.

We present polynomial families complete for the well-studied algebraic complexity classes VF, VBP, VP, and VNP. The polynomial families are based on the homomorphism polynomials studied in the recent works of Durand et al. (2014) and Mahajan et al. (2018). We consider three different variants of graph homomorphisms, namely injective homomorphisms , directed homomorphisms , and injective directed homomorphisms , and obtain polynomial families complete for VF, VBP, VP, and VNP under each one of these. The polynomial families have the following properties: • The polynomial families complete for VF, VBP, and VP are model independent, i.e., they do not use a particular instance of a formula, algebraic branching programs, or circuit for characterising VF, VBP, or VP, respectively. • All the polynomial families are hard under p -projections.

Let$${\mathcal {T}}_\infty (F)$$T∞(F)be the algebra of all$${\mathbb {N}}\times {\mathbb {N}}$$N×Nupper triangular matrices defined over a fieldFof characteristic different from 2. We consider the Jordan triple homomorphisms of$${\mathcal {T}}_\infty (F)$$T∞(F), i.e. the additive maps that satisfy the condition$$\phi (xyx)=\phi (x)\phi (y)\phi (x)$$ϕ(xyx)=ϕ(x)ϕ(y)ϕ(x)for all$$x,y\in {\mathcal {T}}_\infty (F)$$x,y∈T∞(F). For the case whenFis a prime field we find the form of all such maps$$\phi $$ϕ. For the general case we present the form of the surjective maps$$\phi $$ϕ.

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.

In this paper, we first introduce a generalized K-algebra (briefly, gK-algebra) $$(P, \cdot , \odot , e, ^{-1})$$ which is constructed on a non-symmetric polygroup $$(P, \cdot , e, ^{-1})$$ , by adjoining binary hyperoperation $$\odot $$ defined by $$x \odot y = x \cdot y^{-1}$$ , for all $$x, y \in P$$ . Then, we show that every gK-algebra associated to a canonical hypergroup is a hyper B-algebra with constant e. Finally, we define generalized gK-subalgebras, (prime) generalized gK-ideals, homomorphisms of $$(P, \cdot , \odot , e, ^{-1})$$ and quotient generalized gK-algebras.

In this paper, we introduce the notions of $$( \in , \in \vee q^{k}_{\delta })$$ -fuzzy soft near-ring and $$( \in , \in \vee q^{k}_{\delta })$$ -fuzzy soft ideal over a near-ring which are generalizations of $$( \in , \in \vee q_{k})$$ -fuzzy soft near-ring and $$( \in , \in \vee q_{k})$$ -fuzzy soft ideal respectively. We show that the restricted intersection (resp. extended intersection, union, “AND” operation and “OR” operation) of $$( \in , \in \vee q^{k}_{\delta })$$ -fuzzy soft near-rings (resp. ideals) is an $$( \in , \in \vee q^{k}_{\delta })$$ -fuzzy soft near-ring (resp. ideal) over a near-ring. Further, we introduce the concepts of $$( \in , \in \vee q_\delta ^{k})$$ -fuzzy soft subnear-ring and $$( \in , \in \vee q_\delta ^{k})$$ -fuzzy soft ideal of an $$( \in , \in \vee q_\delta ^{k})$$ -fuzzy soft near-ring and provide some related results. Finally, we study the image and pre-image of an $$(\in ,\in \vee q_\delta ^k)$$ -fuzzy soft near-ring (resp. ideal) over a near-ring using fuzzy soft homomorphism.

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.

In the article from the same positions group homomorphisms of matrix groups and ring homomorphisms of matrix rings over associative rings from 1 are described. It is shown that the description of homomorphisms of matrix groups E (n, R) ⊆ G ⊆ GL(n, R), n ≥ 2 into the group of automorphisms GL(W) of the left (optionally free) K-module W over an arbitrary associative ring K of 1 is reduced to cases where 2 or 3 are reversible elements in the ring K. It is proved that they allow a standard description of homomorphisms of the group of elementary transvections $E(n,R), if such a description allows homomorphisms of matrix groups over rings K, in which 2 or 3 are reversible elements. The ring homomorphisms are also described Λ : Rn → End W, n ≥ 2 of the left (optionally free) K-module W over an arbitrary associative ring K of 1. It is shown that homomorphisms Λ allow a standard description on the ring Rn.

Let $$A$$ and $$B$$ be $$C^{*}$$ -algebras, and let $$B$$ be stable. In this paper, we construct an endofunctor $$ \mathfrak R_{X}$$ in the category of $$C^{*}$$ -algebras for every discrete metric space $$X$$ of bounded geometry and introduce the notion of $$ \mathfrak R_{X}$$ -homotopy of $$*$$ -homomorphisms from $$A$$ to $$ \mathfrak R_{X}B$$ . We study the existence problem for a group structure on the set of $$ \mathfrak R_{X}$$ -homotopy classes.

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.

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.

Necessary and sufficient conditions for the exactness (in the algebraic sense) of certain sequences of bounded group homomorphisms are established.

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.

Every homomorphism $\varphi: B(G) \rightarrow B(H)$ between Fourier-Stieltjes algebras on locally compact groups $G$ and $H$ is determined by a continuous mapping $\alpha: Y \rightarrow \Delta(B(G))$, where $Y$ is a set in the open coset ring of $H$ and $\Delta(B(G))$ is the Gelfand spectrum of $B(G)$ (a $*$-semigroup). We exhibit a large collection of maps $\alpha$ for which $\varphi=j_\alpha: B(G) \rightarrow B(H)$ is a completely positive/completely contractive/completely bounded homomorphism and establish converse statements in several instances. For example, we fully characterize all completely positive/completely contractive/completely bounded homomorphisms $\varphi: B(G) \rightarrow B(H)$ when $G$ is a Euclidean- or $p$-adic-motion group. In these cases, our description of the completely positive/completely contractive homomorphisms employs the notion of a "fusion map of a compatible system of homomorphisms/affine maps" and is quite different from the Fourier algebra situation.

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] .

In this paper, we showed how groups are embedded into wreath products, we gave a simpler proof of the theorem by Audu (1991) (see [1]), also proved that a group can be embedded into the wreath product of a factor group by a normal subgroup and also proved that a factor group can be embedded inside a wreath product and the wreath product of a factor group by a factor group can be embedded into a group. We further showed that when the abstract group in the Universal Embedding Theorem is a p-group, cyclic and simple, the embedding becomes an isomorphism. Examples were given to justify the results.

Let M M denote a finitely generated module over a Noetherian ring R R . For an ideal I ⊂ R I \subset R there is a study of the endomorphisms of the local cohomology module H I g ( M ) , g = g r a d e ( I , M ) , H^g_I(M), g = grade(I,M), and related results. Another subject is the study of left derived functors of the I I -adic completion Λ i I ( H I g ( M ) ) \Lambda ^I_i(H^g_I(M)) , motivated by a characterization of Gorenstein rings given in [25]. This provides another Cohen-Macaulay criterion. The results are illustrated by several examples. There is also an extension to the case of homomorphisms of two different local cohomology modules.

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.

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 $$\varphi _n: \Gamma \rightarrow G_n$$ .

Let $ \pi $ be a set of primes. A group $ G $ is weakly $ \pi $ -potent if $ G $ is residually finite and, for each element $ x $ of infinite order in $ G $ , there is a positive integer $ m $ such that, for every positive $ \pi $ -integer $ n $ , there exists a homomorphism of $ G $ onto a finite group which sends $ x $ to an element of order $ mn $ . We obtain a few results about weak $ \pi $ -potency for some groups and generalized free products.

The study of homomorphisms of matrix groups over associative rings began almost 100 years ago with the work of Schreier and Van der Warden and later developed in the works of Dieudonne J., Hua L. K., Reiner I., O’Meara O.T., Hahn A.J., Merzlyakov Yu.I., Waterhouse W.C., Mikhalev O.V., Zelmanov E.I., Golubchik I.Z., Petechuk V.M. and other authors. The study is based on the group properties of a complete linear group GL(n, R) the set of all reversible matrices over the associative ring R with 1. Thus, in all known cases n ≥ 3, despite the difference in the methods used, the automorphisms of the complete linear group were the product of standard automorphisms. It is the reversibility of element 2 that made it possible to consider ever wider classes of rings over which a standard description of homomorphisms of matrix groups is possible. If 2 is an irreversible element, then when n ≥ 3 Petechuk V.M. described the automorphisms of the group GL(n, R) in the case when R is a commutative local ring. It turned out that for n ≥ 4 all automorphisms of such groups are the product of standard automorphisms, and for n = 3 them can be expressed through standard and some non-standard automorphisms. Based on this result, Petechuk V.M. [2] obtained a description of the isomorphisms of the group GL(n, R), n ≥ 3 if R is an arbitrary commutative ring. In particular, he described homomorphisms Λ : P E (n, R) → P GL(m, K), m ≥ 3, n ≥ 3 such ΛP E (n, R) = P H and H ⊇ E (m, K) as over arbitrary commutative rings R and K. From Golubchik I.Z. and Mikhalev O.V. [3], using systems of idempotent, and independently Zelmanov E.I. [4], using the methods of Jordan algebras, obtained a description of the isomorphisms of the group E (n, R), n ≥ 3, 2 ∈ R∗ on the group E (m, K), 2 ∈ K∗ over arbitrary associative rings R and K with 1. Petechuk V.M. [5] described the homomorphisms of a group P E (n, R), n ≥ 3, into a group GL(m, K), m ≥ 2, 2 ∈ K∗ in the case when the fixed submodules of some elements of the fourth order coincide with the fixed submodules of their squares From it follow the results of Golubchik I.Z., Mikhalev О.В. and Zelmanov E.I. Developing techniques related to idempotents, Golubchik I.Z. [6] described the isomorphisms of the groups GL(n, R) and GL(m, K) for n, m ≥ 4 over the associative rings R and K. It turned out that they allow a standard description on the group E (n, R). The authors Petechuk V.M., Yu.V. Petechuk [7, 8] described homomorphisms with condition (*), from which in particular follows the description of isomorphisms of complete linear groups over associative rings. In this paper, methods for describing homomorphisms with the condition (*) are improved and expanded if element 2 is reversible in the ring K and n ≥ 3. The main result of the work is the following theorem. Let R and K be associative rings with 1, 2 ∈ K∗ , E (n, R) ⊆ G ⊆ GL(n, R), n ≥ 3, W be the left Kmodule, homomorphism Λ : G → GL(W) satisfies the condition (*). Then Λ has a standard description on the group E (n, R).

In this paper, the concept of Jordan triple higher -homomorphisms on prime rings is introduced. A result of Herstein is extended on this concept from the ring into the prime ring . We prove that every Jordan triple higher -homomorphism of ring into prime ring is either triple higher -homomorphism or triple higher -anti-homomorphism of into .

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

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 present the notion of perfect ideal of a seminearring S and prove that the kernel of a seminearring homomorphism is a perfect ideal. We show that the quotient structure S/I is isomorphic to the structure $$S_{T(I)}.$$ Finally, we prove isomorphism theorems in seminearrings by using tame condition.

Let q be a power of a prime number. We construct a variety over a finite field F q and show that its middle cohomology realizes the Heisenberg–Weil representations of a general linear group G L n ( q ) for any integer n ≥ 1 . If n is even, by using an endomorphism of this variety, we can construct a representation of a semidirect group G L n ( q ) ⋊ ( Z / 2 Z ) , which is an extension of the Weil representation of G L n ( q ) . By inflating this by a natural homomorphism S p n ( q ) × O 2 + ( q ) → G L n ( q ) ⋊ ( Z / 2 Z ) , we obtain a representation of S p n ( q ) × O 2 + ( q ) . This representation can be used for the Howe correspondence for ( S p n , O 2 + ) in any characteristic. As application, we show that unipotency is preserved under the correspondence for ( S p n , O 2 + ) .

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 $${\mathcal {M}}_\infty (B_{c_0},B_{c_0})$$ .

A homomorphism quasi-order between algebraic structures $\mathcal {A}$ and ${\mathcal{B}}$ of the same type is defined as follows: ${\mathscr{A}}\leq {\mathscr{B}}$ if there is a homomorphism of ${\mathscr{A}}$ to ${\mathscr{B}}$ . The paper deals with the class ${\mathcal{L}}$ of all connected monounary algebras. Let $\sim $ be the equivalence corresponding to the relation ≤ and let $\mathbb {L}={\mathcal{L}}\sim $ . We prove that $\mathbb {L}$ is a lattice. Next it is shown that the lattice $\mathbb {L}$ is distributive. For the purpose of the proof, to each connected monounary algebra without a cycle, an increasing sequence of ordinals is assigned. Finally, we deal with the special case of increasing sequences, namely, such that the sequence contains only positive integers.

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.

Herstein proved that any Jordan homomorphism onto a prime ring of characteristic of different from 2 and 3 is either a homomorphism or an anti-homomorphism. In this paper the concept of Generalized Jordan triple ( )–Higher Homomorphisms (GJT( )-HH) where and are two commuting homomorphisms are introduced as follows: A family of additive mappings of into is said to be a Generalized Triple -Higher Homomorphism (GT( )-HH) if there exist a triple -higher homomorphism (T ( ) – HH) such that for each and for all , we have: and is said to be the relating triple -HH. We will primarily extend the result of Herstein on it. It should be proved that every GJT()-HH of ring into prime ring is either GT( )-HH or triple ( ) higher anti-homomorphism (T( )-HAH).

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 $$\sigma =id_{{\mathcal {R}}}.$$

In this paper, we study local subgraph counting, which is to count the occurrences of a user-given pattern graph p around every node v in a data graph G , when v matches to a given orbit o in p , where the orbit serves as a center to count p. In general, the orbit can be a node, an edge, or a set of nodes in p. Local subgraph counting has played an important role in characterizing high-order local structures that exhibit in a large graph, and has been widely used in denser and relevant communities mining, graphlet degree distribution, discriminative features selection for link prediction, relational classification and recommendation. In the literature, almost all the existing works support a k -node pattern graph, for k ≤ 5, with either 1 node orbit or 1 edge orbit. Their approaches are difficult to support larger k due to the fact that subgraph counting is to count by subgraph isomorphism. In this work, we develop a new general approach to count any k pattern graphs with any orbits selected. The key idea behind is that we do local subgraph counting by homomorphism counting, which can be solved by relational algebra using joins, group-by and aggregation. By homomorphism counting, we do local subgraph counting by eliminating counts for those that are not subgraph isomorphism matchings from the total count for any possible matchings. We have developed a distributed system named DISC on Spark. Our extensive experiments validate the efficiency of our approach by testing 114 local subgraph counting queries used in the existing work over real graphs, where no existing work can support all.

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.