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We are studying topological properties of the Julia set of the map F(z, p) = *** of the complex projective plane PC² to itself. We show a relation between this rational function and an uncountable family of "paper folding" plane filling curves. [ABSTRACT FROM AUTHOR]

An R-moduleM is called δ-weakly Hopfian if any δ-small surjective endomorphism of M is an automorphism. In this paper we explore various properties of δ-weakly Hopfian modules, shedding light on their distinct characteristics. Additionally, we examine the δ-weakly Hopficity of modules over polynomial and truncated polynomial rings. [ABSTRACT FROM AUTHOR]

Important connections in representation theory arise from resolving a finite-dimensional algebra by an endomorphism algebra of a generator-cogenerator with finite global dimension; for instance, Auslander's correspondence, classical Schur–Weyl duality and Soergel's Struktursatz. Here, the module category of the resolution and the module category of the algebra being resolved are linked via an exact functor known as the Schur functor. In this paper, we investigate how to measure the quality of the connection between module categories of (projective) Noetherian algebras, B, and module categories of endomorphism algebras of generator-relative cogenerators over B which are split quasi-hereditary Noetherian algebras. In particular, we are interested in finding, if it exists, the highest degree n so that the endomorphism algebra of a generator-cogenerator provides an n-faithful cover, in the sense of Rouquier, of B. The degree n is known as the Hemmer–Nakano dimension of the standard modules. We prove that, in general, the Hemmer–Nakano dimension of standard modules with respect to a Schur functor from a split highest weight category over a field to the module category of a finite-dimensional algebra B is bounded above by the number of non-isomorphic simple modules of B. We establish methods for reducing computations of Hemmer–Nakano dimensions in the integral setup to computations of Hemmer–Nakano dimensions over finite-dimensional algebras, and vice-versa. In addition, we extend the framework to study Hemmer–Nakano dimensions of arbitrary resolving subcategories. In this setup, we find that the relative dominant dimension over (projective) Noetherian algebras is an important tool in the computation of these degrees, extending the previous work of Fang and Koenig. In particular, this theory allows us to derive results for Schur algebras and the BGG category \mathcal {O} in the integral setup from the finite-dimensional case. More precisely, we use the relative dominant dimension of Schur algebras to completely determine the Hemmer–Nakano dimension of standard modules with respect to Schur functors between module categories of Schur algebras over regular Noetherian rings and module categories of group algebras of symmetric groups over regular Noetherian rings. We exhibit several structural properties of deformations of the blocks of the Bernstein-Gelfand-Gelfand category \mathcal {O} establishing an integral version of Soergel's Struktursatz. We show that deformations of the combinatorial Soergel's functor have better homological properties than the classical one. [ABSTRACT FROM AUTHOR]

We study locally maximal attractors of expanding dynamical systems. Our main result is a representation of these attractors with the help of topological Markov chains corresponding to the Markov partitions of these attractors, which allows us to describe the dynamics of system on them. Ya. G. Sinai was the first who constructed and used Markov partitions for Anosov's diffeomorphisms [Funk. Anal. Prilozh., 2, No 1, 64; No 3, 70 (1968); English translation:Funct. Anal. Appl., 2, No 1, 61; No 3, 245 (1968)]. Expanding endomorphisms regarded as the simplest representatives of endomorphisms were first studied by M. Shub [Amer. J. Math., 91, No 1, 175 (1969)]. To construct Markov partitions for expanding endomorphisms, we update Sinai's approach in the proper way. A more detailed historical overview can be found in the work by O. M. Sharkovsky [Ukr. Mat. Zh., 74, No. 12, 1709 (2023); English translation:Ukr. Math. J., 74, No. 12, 1950 (2023)]. In this work, Sharkovsky indicated that the methods used to prove the main results presented in [Dokl. Akad. Nauk SSSR, 170, No. 6, 1276 (1966); English translation:Sov. Math. Dokl., 7, No. 5, 1384 (1966)] were, in fact, published in the collection of papers "Dynamical systems and the problems of stability of solutions of differential equations" (1973) issued by the Institute of Mathematics of the Academy of Sciences of Ukraine. This collection is difficultly accessible and was never translated into English. Note that, in the indicated paper, these methods were applied to somewhat different objects. To the best of our knowledge, there is no information about publications of similar results. In view of the outlined history and importance of the described approach (based on Markov partitions and topological Markov chains) for the description of construction of the attractors, it seems reasonable to publish these results anew. [ABSTRACT FROM AUTHOR]

This paper studies topological definitions of chain recurrence and shadowing for continuous endomorphisms of topological groups generalizing the relevant concepts for metric spaces. It is proved that in this case the sets of chain recurrent points and chain transitive component of the identity are topological subgroups. Furthermore, we show that some dynamical properties are induced by the original system on quotient spaces. These results link an algebraic property to a dynamical property. [ABSTRACT FROM AUTHOR]

In this paper, we introduce an AEC framework for studying fields with commuting automorphisms. Fields with commuting automorphisms are closely related to difference fields. Some authors define a difference ring (or field) as a ring (or field) together with several commuting endomorphisms, while others only study one endomorphism. Z. Chatzidakis and E. Hrushovski have studied in depth the model theory of ACFA, the model companion of difference fields with one automorphism. Our fields with commuting automorphisms generalize this setting. We have several automorphisms and they are required to commute. Hrushovski has proved that in the case of fields with two or more commuting automorphisms, the existentially closed models do not necessarily form a first order model class. In the present paper, we introduce FCA-classes, an AEC framework for studying the existentially closed models of the theory of fields with commuting automorphisms. We prove that an FCA-class has AP and JEP and thus a monster model, that Galois types coincide with existential types in existentially closed models, that the class is homogeneous, and that there is a version of type amalgamation theorem that allows to combine three types under certain conditions. Finally, we use these results to show that our monster model is a simple homogeneous structure in the sense of S. Buechler and O. Lessman (this is a non-elementary analogue for the classification theoretic notion of a simple first order theory). [ABSTRACT FROM AUTHOR]

In this paper, we give an alternative construction of a certain class of deformed double current algebras. These algebras are deformations of |$ U(\textrm {End}(\Bbbk ^r)[x,y]) $| and they were initially defined and studied by N. Guay in his papers. Here, we construct them as algebras of endomorphisms in Deligne category. We do this by taking an ultraproduct of spherical subalgebras of the extended Cherednik algebras of finite rank. [ABSTRACT FROM AUTHOR]

The main problem of clone theory is to describe the clone lattice for a given basic set. For a two-element basic set this was resolved by E.L. Post, but for at least three-element basic set the full structure of the lattice is still unknown, and the complete description in general is considered to be hopeless. Therefore, it is studied by its substructures and its approximations. One of the possible directions is to examine k-ary parts of the clones and their mutual inclusions. In this paper we study k-ary parts of maximal clones, for k ⩾ 2 , building on the already known results for their unary parts. It turns out that the poset of k-ary parts of maximal clones defined by central relations contains long chains. [ABSTRACT FROM AUTHOR]

In this paper we improve on existing methods to compute quadratic points on modular curves and apply them to successfully find all the quadratic points on all modular curves X_0(N) of genus up to 8, and genus up to 10 with N prime, for which they were previously unknown. The values of N we consider are contained in the set \begin{equation*} \mathcal {L}=\{58, 68, 74, 76, 80, 85, 97, 98, 100, 103, 107, 109, 113, 121, 127 \}. \end{equation*} We obtain that all the non-cuspidal quadratic points on X_0(N) for N\in \mathcal {L} are complex multiplication (CM) points, except for one pair of Galois conjugate points on X_0(103) defined over \mathbb {Q}(\sqrt {2885}). We also compute the j-invariants of the elliptic curves parametrised by these points, and for the CM points determine their geometric endomorphism rings. [ABSTRACT FROM AUTHOR]

Let K be a field of characteristic zero and x a free variable. A K - ℰ -derivation of K [ x ] is a K -linear map of the form I , − , ϕ for some K -algebra endomorphism ϕ of K [ x ] , where I denotes the identity map of K [ x ]. In this paper, we study the image of an ideal of K [ x ] under some K -derivations and K - ℰ -derivations of K [ x ]. We show that the LFED conjecture proposed in [W. Zhao, Some open problems on locally finite or locally nilpotent derivations and ℰ -derivations, Commun. Contem. Math. 20(4) (2018) 1750056] holds for all K - ℰ -derivations and all locally finite K -derivations of K [ x ]. We also show that the LNED conjecture proposed in [W. Zhao, Some open problems on locally finite or locally nilpotent derivations and ℰ -derivations, Commun. Contem. Math. 20(4) (2018) 1750056] holds for all locally nilpotent K -derivations of K [ x ] , and also for all locally nilpotent K - ℰ -derivations of K [ x ] and the ideals u K [ x ] such that either u = 0 , or deg u ≤ 1 , or u has at least one repeated root in the algebraic closure of K. As a bi-product, the homogeneous Mathieu subspaces (Mathieu–Zhao spaces) of the univariate polynomial algebra over an arbitrary field have also been classified. [ABSTRACT FROM AUTHOR]

A ring R is called right monomorphism JP-injective (or MJP-injective for short), if, for any a J(R), every right R-monomorphism from aR to R extends to an endomorphism of R. A ring R is called right monomorphism JGP-injective (or MJGP-injective for short), if, for any 0 = a J(R), there exists a positive integer n such that an = 0 and any right R-monomorphism from anR to R extends to an endomorphism of R. In this paper, several properties of the two classes rings are given. Moreover, some new characterizations of quasi-Frobenius rings are obtained. [ABSTRACT FROM AUTHOR]

An endomorphism of a finite algebra is said to be highest if its pre-period is greater than or equal to the pre-period of all its endomorphisms. In this paper, we characterize all highest endomorphisms of a Boolean lattice. [ABSTRACT FROM AUTHOR]

In this paper the concept of intuitionistic fuzzy quasi injective S-act (for short IFQI-act) and intuitionistic fuzzy injective S-act (IFI-act) have been initiated and some of their properties, also the IF-retract IF-fully invariant and IF-fully stable S-acts have been studied. We show that every IF-retract of IFQI-act is also IFQI-act. Finally, a characterization of IFQI-act has been given in term of the intuitionistic fuzzy endomorphism of the injective envelope of it. Haryeni et al. in [8] found that the edge irregularity strength of fan graphs Fn where n 2 ϵ{2,3,4,5,6} is n + 1. In this paper, we generalise this result for n = 2,3,4, Also we state the edge irregularity strength and modular edge irregularity strength for some lollipop graphs. [ABSTRACT FROM AUTHOR]

Mean dimension may decrease after taking the natural extension. In this paper we show that mean dimension is preserved by natural extension for an endomorphism on a compact metrizable abelian group. As an application, we obtain that the mean dimension of an algebraic cellular automaton coincides with the mean dimension of its natural extension, which strengthens a result of Burguet and Shi [Israel J. Math. (to appear).] with a different proof. [ABSTRACT FROM AUTHOR]

The study of modules by properties of their endomorphisms has long been of interest. In this paper, we introduce the notion of jacobson weakly Hopfian modules. It is shown that over a ring R, every projective (free) R-module is jacobson weakly Hopfian if and only if R has no nonzero semisimple projective R-module. Let L be a module such that L satisfies ascending chain conditions on jacobson-small submodules. Then it is shown that L is jacobson weakly Hopfian. Some basic characterizations of projective jacobson weakly Hopfian modules are proved. [ABSTRACT FROM AUTHOR]

The conference AUTOMATA 2019 was the official annual event of the Technical Committee 1, on Foundations of Computer Science, Working Group 5, on CA and DCS, of the International Federation for Information Processing (IFIP). The present special issue of Natural Computing is dedicated to the study of algebraic, dynamical, algorithmic and complexity-theoretic aspects of cellular automata (CA) and discrete complex systems (DCS). The paper I Complexity-theoretic aspects of expanding cellular automata i by A. Modanese considers expanding cellular automata (XCA), which are one-dimensional CA that can dynamically create new cells between existing ones. [Extracted from the article]

In this paper, we provide the basic properties of (semi)simple hypermodules. We show that if a hypermodule M is simple, then (E n d (M) , ·) is a group, where E n d (M) is the set of all normal endomorphisms of M. We prove that every simple hypermodule is normal projective with a zero singular subhypermodule. We also show that the class of semisimple hypermodules is closed under internal direct sums, factor hypermodules, and subhypermodules. In particular, we give a characterization of internal direct sums of subhypermodules of a hypermodule. [ABSTRACT FROM AUTHOR]

In this paper, we study σ -skew McCoy rings under the σ -compatible or the σ -semicompatible conditions. We show that if R is a semicommutative right or left artinian ring which is σ -semicompatible with an epimorphism σ , then the Jacobson radical J (R) is σ -skew McCoy. As a corollary, we get that the Jacobson radical of a semicommutative artinian ring is right McCoy. We also show that every σ -compatible right duo ring is σ -skew McCoy and that for σ -compatible regular rings, the notions of the σ -skew McCoy and the right McCoy coincide. In addition, we show that every σ -semicompatible semicommutative ring is linearly σ -skew Camillo and that every matrix ring over a division ring is linearly σ -skew Camillo for any endomorphism σ. [ABSTRACT FROM AUTHOR]

In this paper, we properly generalize the notion of co-Hopficity for modules to the concept of closed co-Hopficity. A module M is said to be closed co-Hopfian if any injective endomorphism of M has a closed submodule image. The aim of this paper is to study and investigate this class of modules. In addition, some relations for this class with other types of modules are provided. [ABSTRACT FROM AUTHOR]

It is expected that a totally invariant divisor of a non-isomorphic endomorphism of the complex projective space is a union of hyperplanes. In this paper, we compute an upper bound for the degree of such a divisor. As a consequence, we prove the linearity of totally invariant divisors with isolated singularities. [ABSTRACT FROM AUTHOR]

In this paper, we study the semi-symmetry and pseudo-symmetry of almost cosymplectic 3 -manifolds. First, we prove that an H -almost cosymplectic 3 -manifold M is semi-symmetric if and only if it is cosymplectic. Here by an H -almost cosymplectic 3 -manifold, we mean an almost cosymplectic 3 -manifold whose characteristic vector field ξ is a harmonic unit vector field. If an almost cosymplectic 3 -manifold M whose fundamental endomorphism field h is parallel in the direction of the characteristic vector field ξ ( ∇ ξ h = 0), then it is H -almost cosymplectic. In particular, an almost cosymplectic 3 -manifold M satisfying ∇ ξ h = 0 is semi-symmetric if and only if it is cosymplectic. Next, we prove that pseudo-symmetric H -almost cosymplectic 3 -manifolds are certain generalized almost cosymplectic (κ , μ , ν) -spaces. [ABSTRACT FROM AUTHOR]

Over fields of characteristic 2, Specht modules may decompose and there is no upper bound for the dimension of their endomorphism algebra. A classification of the (in)decomposable Specht modules and a closed formula for the dimension of their endomorphism algebra remain two important open problems in the area. In this paper, we introduce a novel description of the endomorphism algebra of the Specht modules and provide infinite families of Specht modules with one-dimensional endomorphism algebra. [ABSTRACT FROM AUTHOR]

In this paper, we first show that the monoid of separable surjective self-morphisms of a variety of Ueno type coincides with the group of automorphisms. We also give an explicit description of the automorphism group. As applications, we confirm Kawaguchi–Silverman conjecture for automorphisms of a variety of Ueno type and some Calabi–Yau three-fold, defined over ℚ¯. [ABSTRACT FROM AUTHOR]

Important correspondences in representation theory can be regarded as restrictions of the Morita–Tachikawa correspondence. Moreover, this correspondence motivates the study of many classes of algebras like Morita algebras and gendo‐symmetric algebras. Explicitly, the Morita–Tachikawa correspondence describes that endomorphism algebras of generators–cogenerators over finite‐dimensional algebras are exactly the finite‐dimensional algebras with dominant dimension at least two. In this paper, we introduce the concepts of quasi‐generators and quasi‐cogenerators that generalise generators and cogenerators, respectively. Using these new concepts, we present higher versions of the Morita–Tachikawa correspondence that take into account relative dominant dimension with respect to a self‐orthogonal module with arbitrary projective and injective dimensions. These new versions also hold over Noetherian algebras that are finitely generated and projective over a commutative Noetherian ring. [ABSTRACT FROM AUTHOR]

AbstractIn this paper we consider the monoids of all partial endomorphisms, of all partial weak endomorphisms, of all injective partial endomorphisms, of all partial strong endomorphisms and of all partial strong weak endomorphisms of a star graph with a finite number of vertices. Our main objective is to determine their ranks. We also describe their Green’s relations, calculate their cardinalities and study their regularity. [ABSTRACT FROM AUTHOR]

Let r and s be positive integers such that r ⩾ 3. Let U 1 , ... , U r be vector spaces over a field F and V 1 , ... , V s be vector spaces over a field K such that dim ⁡ U i , dim ⁡ V j ⩾ 2 for all i , j. In this paper, we characterize maps ψ : ⨂ i = 1 r U i → ⨂ i = 1 s V i that preserve adjacency in both directions, which extends Hua's fundamental theorem of geometry of rectangular matrices. We also characterize related results concerning locally full maps preserving adjacency in both directions between tensor spaces, maps preserving adjacency in both directions between tensor spaces over a field all whose nonzero endomorphisms are automorphisms, and injective continuous adjacency preserving maps on finite dimensional tensor spaces over the real field. [ABSTRACT FROM AUTHOR]

AbstractLet V be a vector space and U a fixed subspace of V. We denote the semigroup of all linear transformations on V under composition of functions by L(V). In this paper, we study the semigroup of all linear transformations on V whose restrictions belong to the general linear group GL(U), denoted by LGL(U)(V). More precisely, we consider the subsemigroup LGL(U)(V)={α∈L(V):α|U∈GL(U)} of L(V). In this work, Green’s relations and ideals of this semigroup are described. Then we also determine the minimal ideal and the set of all minimal idempotents of it. Moreover, we establish an isomorphism theorem when V is a finite dimensional vector space over a finite field. Finally, we find its generating set. [ABSTRACT FROM AUTHOR]

In the past few decades, researchers have found that studying modules using endomorphisms is a powerful and useful tool. This has led to valuable works in this field. Recently, the study of (dual) Rickart modules has become an important approach as they are deeply connected to endomorphisms. Building on this work, the authors introduce a new perspective on (dual) Rickart modules using isomorphism. We also define virtually (dual) Rickart modules. It is shown that rings with all modules virtually Rickart are semisimple rings. The paper includes various examples to illustrate the concepts presented. [ABSTRACT FROM AUTHOR]

In this paper, we study linear codes invariant under a cyclic endomorphism T , called T -cyclic codes. Since every cyclic endomorphism can be represented by a cyclic matrix with respect to a given basis, all of these matrices are similar. For simplicity we restrict ourselves to linear codes invariant under the right multiplication by a cyclic matrix M , that we call M -cyclic codes, and when M is the companion matrix C f of a given nonzero polynomial f (x) we call them f -cyclic codes. The similarity relation between matrices helps us find connections between M -cyclic codes and f -cyclic codes, where f (x) is the minimal polynomial of M. The class of M -cyclic codes contains cyclic codes and their various generalizations such as constacyclic codes, right and left polycyclic codes, monomial codes, and others. As common in the study of cyclic codes and their generalizations, we make use of the one-to-one correspondence between M -cyclic codes and ideals of the polynomial ring R f : = q [ x ] / 〈 f (x) 〉 , where f (x) is the minimal polynomial of M. This correspondence leads to some basic characterizations of these codes such as generator and parity check polynomials among others. Next, we study the duality of these codes, where we show that the b -dual of an M -cyclic code is an M ∗ -cyclic code, where M ∗ is the adjoint matrix of M with respect to b , and we explore some important results on the duality of these codes. Finally, we give examples as applications of some of the results and we construct some optimal codes. [ABSTRACT FROM AUTHOR]

In this paper, we introduce an algebra structure denoted by InvDer algebra in which we twist an algebra with an invertible derivation whose inverse is also a derivation. We define InvDer Lie algebras, InvDer associative algebras, InvDer Zinbiel algebras and InvDer dendriforme algebras. We also study the relations between these structures by using Rota–Baxter operators and endomorphism operators. [ABSTRACT FROM AUTHOR]

Despite tangible advances in machine learning (ML) over the last few decades, many of the ML techniques still suffer from fundamental issues like overfitting and lack of explainability. These issues mandate requirements for mathematical rigor to ensure robust learning from observed data. In this context, topological invariants in data manifolds provide a rich representation of the underlying dynamical system, which can be utilized for developing a mathematically rigorous ML tool to characterize the dynamical behaviour and operational phases of the underlying process. This paper aims to investigate spectral invariants of symbolic systems for detecting changes in topological characteristics of data manifolds. A novel ML approach is proposed, where commutator norms are used on sequences of endomorphisms to symbolically describe dynamical systems on probability spaces with ergodic measures. The objective here is to detect topological invariants of data manifolds that can be used for signal processing, pattern recognition, and anomaly detection. The proposed ML approach is validated on models of selected chaotic dynamical systems for prompt detection of phase transitions. This article is part of the theme issue 'Data-driven prediction in dynamical systems'. [ABSTRACT FROM AUTHOR]

In this paper we introduce dual F -Baer modules and give a characterization of them. Let M be a module and F a fully invariant submodule of M. M is called dual F -Baer if for every family of endomorphisms { g α } α ∈ I of M , ∑ α ∈ I g α (F) is a direct summand of M. We prove that M is dual F -Baer if and only if M = F ⊕ N for some submodule N of M with F dual Baer. We obtain a positive solution for the Schröder–Bernstein problem for certain dual F -Baer modules. [ABSTRACT FROM AUTHOR]

Let k be an arbitrary field, Λ be a k-algebra and V be a Λ -module. When it exists, the universal deformation ring R (Λ , V) of V is a k-algebra whose local homomorphisms to R parametrize the lifts of V up to R ⊗ k Λ , where R is any complete, local commutative Noetherian k-algebra with residue field k. Symmetric special biserial algebras, which coincide with Brauer graph algebras, can be viewed as generalizing the blocks of finite type p-modular group algebras. Bleher and Wackwitz classified the universal deformation rings for all modules for symmetric special biserial algebras with finite representation type. In this paper, we begin to address the tame case. Specifically, let Λ be any 1-domestic, symmetric special biserial algebra. By viewing Λ as generalized Brauer tree algebras and making use of a derived equivalence, we classify the universal deformation rings for those Λ -modules V with stable endomorphism ring isomorphic to k. The latter is a natural condition, since it guarantees the existence of the universal deformation ring R (Λ , V) . [ABSTRACT FROM AUTHOR]

The main goal of this paper is to introduce analogously modal operators on L-algebras and study their properties. To begin with, we introduce the notion of modal operators on L-algebras and investigate some important properties of this operator. In order for the kernel of modal operator to be ideal, we investigate what conditions are required. Relations between modal operator and endomorphism of L-algebras are investigated. Also, we define the concept of positive L-algebra and some characterizations of positive L-algebra are established. Finally, we introduce a map ka and show that ka is a modal operator and we prove that the set of all ka on a positive L-algebra makes a dual BCK-algebra. [ABSTRACT FROM AUTHOR]

There are six classes of endomorphisms for a graph. The sets of these endomorphisms form a chain under the inclusion of sets. In order to systematically study these endomorphisms, Böttcher and Knauer defined the concepts of the endomorphism spectrum and endomorphism type of a graph in 1992. In this paper, based on the property and structure of the endomorphism monoids of graphs, six classes of endomorphisms of double-edge fan graphs are described. In particular, we give the endomorphism spectra and endomorphism types of double-edge fan graphs. [ABSTRACT FROM AUTHOR]

This paper studies Rota-Baxter (RB) operators in complex semi-simple algebras A. They are certain C-linear endomorphisms of A, when considered the latter as a C-vector space. Properties as the nilpotency or the spectrum of such operator R are studied. Some examples are given when A is a Clifford algebra. [ABSTRACT FROM AUTHOR]

The first goal of this paper is to investigate the Pierce decomposition of the endomorphism ring End ⁡ (G) = F ^ ⊕ End s ⁡ (G) of an abelian p-group G and its application to the recent studies of groups with minimal full inertia and of thick-thin groups. The second goal is to investigate the Pierce embedding Ψ : End ⁡ (G) / H ⁢ (G) → ∏ n M f n ⁢ (G) . We prove that more classes of groups than those described by Pierce have the property that the map Ψ is surjective, and we furnish examples of groups which do not have this property. Several results connecting the Pierce decomposition and the Pierce embedding of End ⁡ (G) are obtained that allow one to derive general conditions on a group G which ensure that the Pierce embedding of End ⁡ (G) is not surjective. [ABSTRACT FROM AUTHOR]

In one of our former papers Endomorphisms of the measure algebra of commutative hypergroups we considered exponential monomials on hypergroups and higher order derivations of the corresponding measure algebra. Continuing with this, we are now looking for the connection between the generalized exponential polynomials of a commutative hypergroup and the higher order derivations of the corresponding measure algebra. [ABSTRACT FROM AUTHOR]

There are six different classes of endomorphisms for a graph. The sets of these endomorphisms always form a chain under the inclusion of sets. In order to study these different endomorphisms more systematically, Böttcher and Knauer proposed the concept of the endomorphism type of a graph in 1992. In this paper, we explore the six different classes of endomorphisms of graph P (3 m + 1 , 3) . In particular, the endomorphism type of P (3 m + 1 , 3) is given. [ABSTRACT FROM AUTHOR]

In the paper, we study the semiring of skew polynomials over a Rickart Bezout semiring. Namely, let every left annihilator ideal of a semiring be an ideal. Then the semiring of skew polynomials is a semiring without nilpotent elements, and every its finitely generated left monic ideal is principal if and only if is a left Rickart left Bezout semiring, is a rigid endomorphism, and is invertible for any nonzerodivisor . We also obtain a characterization of the semiring in terms of Pierce stalks of the semiring . The structure of left monic ideals of the semiring of skew polynomials over a left Rickart left Bezout semiring is clarified. [ABSTRACT FROM AUTHOR]

Let S be a semigroup, and let φ, ψ: S → S be two endomorphisms (which are not necessarily involutive). Our main goal in this paper is to solve the following generalized variant of d'Alembert's functional equation f (x ϕ (y)) + f (ψ (y) x) = 2 f (x) f (y) , x , y ∈ S , where f : S → ℂ is the unknown function by expressing its solutions in terms of multiplicative functions. Some consequences of this result are presented. [ABSTRACT FROM AUTHOR]

Let be a class of Abelian groups. A group is said to be determined by its endomorphism semigroup in the class if every isomorphism , where , implies the isomorphism . The paper describes those Abelian groups in the class of completely decomposable quotient divisible Abelian groups which are determined by their endomorphism semigroups in the class . [ABSTRACT FROM AUTHOR]

In this paper, local unstable metric entropy, local unstable topological entropy and local unstable pressure for partially hyperbolic endomorphisms are introduced and investigated. Specially, two variational principles concerning relationships among the above mentioned numbers are formulated. [ABSTRACT FROM AUTHOR]