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In this paper, we generalize some results from [M. Ashraf and S. Ali, On left multipliers and the commutativity of prime rings, Demonstr. Math. 41 2008, 4, 763–771] and present our contribution to the study of endomorphisms. In particular, we prove that a prime ring must be an integral domain if it admits endomorphisms and multipliers satisfying some algebraic identities on non-zero ideals. As a consequence of our main theorems, many known results can either be generalized or deduced. Furthermore, an example is given to show that the necessity of the primeness assumption imposed on the hypotheses of various theorems cannot be unnecessary. [ABSTRACT FROM AUTHOR]

We describe injective endomorphisms of the inverse semigroup B ω F introduced in [Visn. Lviv. Univ., Ser. Mech.-Mat., Issue 90, 5 (2020)] for the case where the family F n is generated by the set {0,1,..., n}. In particular, it is shown that the semigroup of injective endomorphisms of the semigroup B ω F is isomorphic to (ω, +). Moreover, we describe the structure of the semigroup E n d B λ of all endomorphisms of the semigroup of λ × λ -matrix units B λ . [ABSTRACT FROM AUTHOR]

Let 휅 be an inaccessible cardinal, 픘 a universal algebra, and ∼ the equivalence relation on U κ of eventual equality. From mild assumptions on 휅, we give general constructions of E ∈ End (U κ / ∼) satisfying E ∘ E = E which do not descend from Δ ∈ End ⁡ (U κ) having small strong supports. As an application, there exists an E ∈ End (Z κ / ∼) which does not come from a Δ ∈ End ⁡ (Z κ) . [ABSTRACT FROM AUTHOR]

We investigate the class of endomorphisms α of a ring U T M n (R) of upper triangular n × n matrices such that α (e ij) is a (0,1)-matrix for any matrix unit e ij . We use the left and right semicentral idempotents defined and studied by Birkenmeier. We study the idempotent semigroup (E n (R) ,.) of endomorphisms of U T M n (R) . An endomorphism α is called regular if α (e ii) = e ii or α (e ii) = 0 for all i = 1 , ... , n . In the main results we prove that the class of regular (0,1)-endomorphisms is E n (R) , that the semigroup (E n (R) ,.) consists of all idempotent (0,1)-endomorphisms and all other (0,1)-endomorphisms are roots of idempotents. [ABSTRACT FROM AUTHOR]

Equivalence relations or, more general, quasiorders (i.e., reflexive and transitive binary relations) ϱ have the property that an n-ary operation f preserves ϱ , i.e., f is a polymorphism of ϱ , if and only if each translation (i.e., unary polynomial function obtained from f by substituting constants) preserves ϱ , i.e., it is an endomorphism of ϱ. We introduce a wider class of relations—called generalized quasiorders—of arbitrary arities with the same property. With these generalized quasiorders we can characterize all algebras whose clone of term operations is determined by its translations by the above property, what generalizes affine complete algebras. The results are based on the characterization of so-called u-closed monoids (i.e., the unary parts of clones with the above property) as Galois closures of the Galois connection End – gQuord , i.e., as endomorphism monoids of generalized quasiorders. The minimal u-closed monoids are described explicitly. [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]

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We consider linear isotopes of commutative groups, i.e., central quasigroups and study the invertibility and orthogonality conditions. It turns out that it is sufficient to study these conditions solely for the unitary isotopes, i.e., for isotopes, which have an idempotent. We establish criteria for the possession of each invertibility property (inverse property, crossed inverse property, and mirroring) for unitary central and matrix quasigroups. In particular, for matrices of the second order, we describe the corresponding matrix quasigroups over the fields of characteristics 2 and 3. The orthogonality criteria are obtained for matrix quasigroups with the indicated invertibility properties. [ABSTRACT FROM AUTHOR]

We show that robustly transitive endomorphisms of a closed manifold must have a non-trivial dominated splitting or be a local diffeomorphism. This allows to get some topological obstructions for the existence of robustly transitive endomorphisms. To obtain the result, we must understand the structure of the kernel of the differential and the recurrence to the critical set of the endomorphism after perturbation. [ABSTRACT FROM AUTHOR]

We prove some basic results on left groups and their homomorphisms and characterize congruences and homomorphisms on these semigroups. Finally we investigate some properties of the endomorphism monoid of a left group. [ABSTRACT FROM AUTHOR]

A graph is a core or unretractive if all its endomorphisms are automorphisms. Well-known examples of cores include the Petersen graph and the graph of the dodecahedron—both generalized Petersen graphs. We characterize the generalized Petersen graphs that are cores. A simple characterization of endomorphism-transitive generalized Petersen graphs follows. This extends the characterization of vertex-transitive generalized Petersen graphs due to Frucht, Graver, and Watkins and solves a problem of Fan and Xie. Moreover, we study generalized Petersen graphs that are (underlying graphs of) Cayley graphs of monoids. We show that this is the case for the Petersen graph, answering a recent mathoverflow question, for the Desargues graphs, and for the Dodecahedron—answering a question of Knauer and Knauer. Moreover, we characterize the infinite family of generalized Petersen graphs that are Cayley graphs of a monoid with generating connection set of size two. This extends Nedela and Škoviera's characterization of generalized Petersen graphs that are group Cayley graphs and complements results of Hao, Gao, and Luo. [ABSTRACT FROM AUTHOR]

This article describes all injective endomorphisms of the classical Toeplitz algebra. Their connection with endomorphisms of the algebra of continuous functions on the unit circle and with coverings over the unit circle was considered. It was shown that each non-unitary isometry V in the Toeplitz algebra determines the identity preserving endomorphism, as well as the class of its compact perturbations, i.e., identity non-preserving endomorphisms, defined by partial isometries {V P}, where P is a projection of finite codimension. The notions of T -equivalence of endomorphisms and T -equivalence up to a compact perturbation were introduced. An example was provided wherein the isometries are unitarily equivalent but the corresponding endomorphisms fall into different equivalence classes. Of all endomorphisms, the ones belonging to the class of Blaschke endomorphisms, which are analogous to endomorphisms of the discalgebra and generate unbranched coverings over the unit circle, were singled out.

We study the endomorphisms and derivations of an infinite-dimensional cyclic Leibniz algebra. Among others it was found that if L is a cyclic infinite-dimensional Leibniz algebra over a field F , then the group of all automorphisms of L is isomorphic to a multiplicative group of the field F. The description of an algebra of derivations of a cyclic infinite-dimensional Leibniz algebra has been obtained. [ABSTRACT FROM AUTHOR]

László Fuchs posed the following question: which abelian groups arise as the group of units in a ring? In this paper, we investigate a related question: for such realizable groups G , when is there a ring R with unit group G such that every group endomorphism of G is induced by a ring endomorphism of R ? We answer this question for four common classes of groups: torsion-free abelian groups, groups of odd order, torsion abelian groups, and finitely generated abelian groups. [ABSTRACT FROM AUTHOR]

Let (S, +) be an abelian semigroup, let (H, +) be an abelian group which is uniquely 2-divisible, and let ϕ be an endomorphism of S. We find the solutions f, h : S → H of each of the functional equations f(x+y)+f(x+ϕ(y))=h(x)+f(y)+f∘ϕ(y), x,y∈S,f(x+y)+f(x+ϕ(y))=h(x)+2f(y), x,y∈S,\matrix{ {f\left( {x + y} \right) + f\left( {x + \varphi \left( y \right)} \right) = h\left( x \right) + f\left( y \right) + f \circ \varphi \left( y \right),\,x,y \in S,} \hfill \cr {f\left( {x + y} \right) + f\left( {x + \varphi \left( y \right)} \right) = h\left( x \right) + 2f\left( y \right),\,x,y \in S,} \hfill \cr } in terms of additive and bi-additive maps. Moreover, as applications, we determine the solutions of some related functional equations.

This article focuses on decomposing a Banach algebra X via its endomorphisms. In particular, we show that if a Banach algebra X has an injective continuous endomorphism f such that f([xn; ym]) is in the center of X, Z(X), for two integers n = n(x; y);m = m(x; y) and sufficiently many x; y, then for all x in X either x in Z(X) or f(x) in Z(X). To demonstrate the importance of our theorem assumptions, we will provide several examples. [ABSTRACT FROM AUTHOR]

Let ℧ be a prime ring of char(℧) ≠ 2 with its center Z. This article introduces new classes of endomorphisms and investigates how they relate to antiautomorphisms of prime rings and the commutativity of prime rings. Additionally, we fully describe and classify some of these endomorphisms. We also give examples to prove the necessity of the numerous restrictions included in the hypotheses of our results. [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]

The aim of the paper is to introduce a new class of rings — the inner generalized Weyl algebras (IGWA) — and to give simplicity criteria for them. For each IGWA A a derivative series of IGWAs, A → A′→ A″ →⋯ → A(α) →⋯, is attached where α is an arbitrary ordinal. In general, all rings A(α) are distinct. A new construction of rings, the inner (σ,τ,a)-extension of a ring, is introduced (where σ and τ are endomorphisms of a ring D and a ∈ D). [ABSTRACT FROM AUTHOR]

The notion of a pre-period of an algebra A is defined by means of the notion of the pre-period λ(f) of a monounary algebra hA; fi: it is determined by sup{λ(f) | f is an endomorphism of A}. In this paper we focus on the pre-period of a finite modular lattice. The main result is that the pre-period of any finite modular lattice is less than or equal to the length of the lattice; also, necessary and sufficient conditions under which the pre-period of the glued sum is equal to the length of the lattice, are shown. Moreover, we show the triangle inequality of the pre-period of the glued sum. [ABSTRACT FROM AUTHOR]

We characterize the sets of homomorphisms, endomorphisms and automorphisms of n-ary groups with cyclic retracts. [ABSTRACT FROM AUTHOR]

Some methods have been used to express a finitely generated module over a principal ideal domain as a finite direct sum of its cyclic submodules. In this paper, we give an alternative technique to decompose a free module with finite rank over a principal ideal domain using eigen spaces of its endomorphism ring. [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]

Let S be a semigroup, let H be a 2 -torsion free abelian group, and let φ , ψ : S → S be two endomorphisms that need not be involutive. We express the solutions f : S → H of generalized variant of Jensen's functional equation \begin{equation*} f(x\varphi(y))+f(\psi(y)x)=2f(x), \quad x,y\in S, \end{equation*} in terms of additive maps. In all results it is supposed that at least one of the endomorphisms φ and ψ is surjective. Many consequences of this result are presented. [ABSTRACT FROM AUTHOR]

We prove the endomorphism conjecture for graded posets with largest Whitney number at most 4. [ABSTRACT FROM AUTHOR]

The set of all endomorphisms over -module is a non-empty set denoted by . From we can construct the ring of over addition and composition function. The prime ideal is an ideal which satisfies the properties like the prime numbers. In this paper, we take the ring of integer number and the module of over such that the is a ring. Furthermore, we show the existences of prime ideal on the . We also applied a prime ideal property to prime ideal on .

In this paper, we show a way to characterize the R-automorphisms of formal power series on several indeterminates and with coefficients over a commutative ring with identity, R. We show this characterization, as an extension of existing result for the R-automorphisms of the formal power series in an indeterminate, given by O’Malley, M. and Wood, C. in [12].

Let P be a left LCM semigroup, and α an action of P by endomorphisms of a C*-algebra A. We study a semigroup crossed product C*-algebra in which the action α is implemented by partial isometries. This crossed product gives a model for the Nica-Teoplitz algebras of product systems of Hilbert bimodules (associated with semigroup dynamical systems) studied first by Fowler, for which, we provide a structure theorem as it behaves well under short exact sequences and tensor products. [ABSTRACT FROM AUTHOR]

In this paper, we show that smooth TA-endomorphisms of compact manifolds with c-expansivity (that is, expansive in the inverse limit) and C1-stable shadowing are Axiom A. [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(xϕ(y))+f(ψ(y)x)=2f(x)f(y), x,y ∈ S,f\left( {x\varphi \left( y \right)} \right) + f\left( {\psi \left( y \right)x} \right) = 2f\left( x \right)f\left( y \right),\,\,\,\,\,\,x,y\, \in \,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.

Let ℧ be a prime ring of char(℧) ≠2 with its center Z. This article introduces new classes of endomorphisms and investigates how they relate to antiautomorphisms of prime rings and the commutativity of prime rings. Additionally, we fully describe and classify some of these endomorphisms. We also give examples to prove the necessity of the numerous restrictions included in the hypotheses of our results.

The crucial step in elliptic curve scalar multiplication based on scalar decompositions using efficient endomorphisms—such as GLV, GLS or GLV+GLS—is to produce a short basis of a lattice involving the eigenvalues of the endomorphisms, which usually is obtained by lattice basis reduction algorithms or even more specialized algorithms. Recently, lattice basis reduction is found to be unnecessary. Benjamin Smith (AMS 2015) was able to immediately write down a short basis of the lattice for the GLV, GLS, GLV+GLS of quadratic twists using elementary facts about quadratic rings. Certainly it is always more convenient to use a ready-made short basis than to compute a new one by some algorithm. In this paper, we extend Smith's method on GLV+GLS for quadratic twists to quartic and sextic twists, and give ready-made short bases for 4-dimensional decompositions on these high degree twisted curves. In particular, our method gives a unified short basis compared with Hu et al.'s method (DCC 2012) for 4-dimensional decompositions on sextic twisted curves.

We introduce the notion of intrinsic semilattice entropy h ̃ in the category ℒ qm of generalized quasimetric semilattices and contractive homomorphisms. By using appropriate categories and functors F : → ℒ qm , we find specific known entropies h ̃ on as intrinsic functorial entropies, that is, as h ̃ = h ̃ ∘ F. These entropies are the intrinsic algebraic entropy, the algebraic and the topological entropies for locally linearly compact vector spaces, the topological entropy for totally disconnected locally compact groups and the algebraic entropy for compactly covered locally compact abelian groups. [ABSTRACT FROM AUTHOR]

Based on the previously obtained concrete characterization of the endomorphism semigroups of quasi-acyclic reflexive graphs we prove the relatively elementary definability of the class of such graphs in the class of all semigroups. It will permit us to investigate for such graphs the abstract representation problem for the endomorphism semigroups of graphs and the problem of elementary definability of graphs by their endomorphism semigroups. [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.

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(3m+1,3). In particular, the endomorphism type of P(3m+1,3) is given.

We investigate a generalization of many functional equations. Namely, we consider the following functional equation ∫S f(x + y + t) dμ(t) + ∫S f(x + φ(y) + t) dν(t) = f(x) + h(y), x, y ∈ S, where (S,+) is an abelian semigroup, φ is a surjective endomorphism of S, E is a linear space over the field K ∈ {R, C} and μ,ν are linear combinations of Dirac measures. Under appropriate conditions on μ and ν and based on Stetkar's result [9], we find and characterize solutions of the previous functional equation. [ABSTRACT FROM AUTHOR]

In the paper we examined the some effects of derivation, trace of permuting tri-derivation and endomorphism on each other in prime and semiprime ring. Let R be a 2, 3-torsion free prime ring and F : R x R x R → R be a permuting tri-derivation with trace f, d : R → R be a derivation. In particular, the following assertions have been proved: • if [d(r), r] = f(r) for all r • R, then R is commutative or d = 0 (Theorem 1); • if g : R → R is an endomorphism such that F(d(r), r, r) = g(r) for all r • R, then F = 0 or d = 0 (Theorem 2); • if F(d(r), r, r) = f(r) for all r • R, then (i) F = 0 or d = 0, (ii) d(r) • f(r) = 0 for all r • R (Theorem 3). In the other hand, if there exist permuting tri-derivations F1, F2 : R x R x R → R such that F1(f2(r), r, r) = f1(r) for all r • R, where f1 and f2 are traces of F1 and F2, respectively, then (i) F1 = 0 or F2 = 0, (ii) f1(r) • f2(r) = 0 for all r • R (Theorem 4). [ABSTRACT FROM AUTHOR]

The pre-period of a lattice (resp. pre-period of a lattice fixing α) A is defined as the supremum of the pre-period of an endomorphism (resp. a connected endomorphism fixing α) of A and is denoted by λ(A) (resp. λα(A)). Let MPP (resp. NPP) be the set of all finite modular lattices A with λ(A) = ℓ(A) (resp. λ(A) = max {λ0(A), λ1(A)},) where ℓ(A) is the length of A. We will show that MPP (NPP (M, where M is the set of all finite modular lattices. [ABSTRACT FROM AUTHOR]

We introduce the notions of outside, inside, (r, l)- and (l, r)-derivations induced by an endomorphism of BG-algebras and give some properties. [ABSTRACT FROM AUTHOR]

In this work, we study the endomorphism set of quasi-injective gamma acts, by defining some concepts like J(M), T(M) for any SΓ-act M, and the important result we predict the regularity properties' of endomorphism set under some conditions and we study a set where x is an element of M we give description of in terms of the endomorphism α – monoid of quasi-injective gamma acts. [ABSTRACT FROM AUTHOR]

The concept of an α - nil reversible ring is a generalization of α - reversible ring as well as an extension of nil reversible rings. We first consider basic properties of α-nil reversible rings. Then we investigate extensions of α-nil reversible, including trivial extension, Dorroh extension and Jordan extension. [ABSTRACT FROM AUTHOR]

An algebra $\cal A$ is said to have the endomorphism kernel property (EKP) if every congruence on $\cal A$ is the kernel of some endomorphism of $\cal A$. Three classes of monounary algebras are dealt with. For these classes, all monounary algebras with EKP are described.

Let X be a normal Q-factorial projective variety with at most log canonical singularities. We shall give a sufficient condition for the existence of at most finitely many KX-negative extremal rays R(C NE(X) of divisorial type. As an application, we show that for a nonisomorphic surjective endomorphism f:X - X of a normal projective Q-factorial terminal 3-fold X with (X) > 0, a suitable power fk (k > 0) of f descends to a nonisomorphic surjective endomorphism g:Xmin - Xmin of a minimal model Xmin of X. [ABSTRACT FROM AUTHOR]

We classify all partial equivalence relations according to their endotype with respect to endotopisms and compute the cardinalities of endotopism semigroups (strong endotopism monoids, autotopism groups) of partial equivalence relations on a finite set. [ABSTRACT FROM AUTHOR]

A graph G is called End-regular, if every endomorphism of G is regular as a monoid. In this article, we investigate End-regularity of bicycle graphs. Moreover, a generalization of bicycle graph is defined and gives a characterization of End-regular generalized bicycle graphs. Furthermore, properties End-idempotent-closed and End-orthodox have been considered for these graphs. Finally, it has been proved that when join of two generalized bicycle graphs is End-regular.