Set Of Endomorphisms Research Articles

Counting points of bounded height in monoid orbits

Given a set of endomorphisms on $$\mathbb {P}^N$$ , we establish an upper bound on the number of points of bounded height in the associated monoid orbits. Moreover, we give a more refined estimate with an associated lower bound when the monoid is free. Finally, we show that most sets of rational functions in one variable satisfy these more refined bounds.

Blenders near polynomial product maps of $\mathbb{C}^2$

In this paper we show that if p is a polynomial of degree d\geq2 possessing a neutral periodic point then a product map of the form (z,w)\mapsto(p(z),q(w)) can be approximated by polynomial skew products (z,w)\mapsto(\tilde{p}(z,w),q(w)) possessing special dynamical objects called blenders. Moreover, these objects can be chosen to be of two types: repelling or saddle. As a consequence, such a product map belongs to the closure of the interior of two different sets: the bifurcation locus of the space of holomorphic endomorphisms of degree d of \mathbb{P}^2 and the set of endomorphisms having an attracting set of non-empty interior. Similar techniques also give the first example of an attractor with non-empty interior or of a saddle hyperbolic set which is robustly contained in the small Julia set and whose unstable manifolds are all dense in \mathbb{P}^2 . In an independent part, we use perturbations of Hénon maps to obtain examples of attracting sets with repelling points and also of quasi-attractors which are not attracting sets.

Projections on right and left ideals of endomorphism semiring which are derivations

The aim of this paper is the investigation of the derivations in an endomorphism semiring of a finite chain. Such semiring can be represented as a simplex and its subsimplices are left ideals of the semiring. We construct projections on these left ideals and prove that they are derivations and also find the maximal subsemirings of the simplex which are the domains of the constructed derivations. Consequently, we obtain some results concerning nilpotent endomorphisms and using well-known result of Stanley we prove that order of semiring of nilpotent endomorphisms is equal to [Formula: see text], where [Formula: see text] is the [Formula: see text]th Catalan number. We consider a class of right ideals of the semiring and introduce projections on these ideals which are derivations and also find the maximal subsemirings of the simplex which are the domains of the constructed derivations. For one of these derivations [Formula: see text] and for a fixed endomorphism [Formula: see text] of a considered right ideal, the set of endomorphisms [Formula: see text] such that [Formula: see text] is denoted by [Formula: see text]. The last set is a semiring if and only if [Formula: see text] is an idempotent. The number of the semirings [Formula: see text], where [Formula: see text], is equal to [Formula: see text], which is the [Formula: see text]th Fibonacci number.

QUASI-INJECTIVE NEAR-RING GROUPS

We extend the concepts of quasi-injective modules and their en- domorphism rings to near-ring groups. We attempt to derive the near-ring character of the set of endomorphism of quasi-injective N-groups under certain conditions and this leads us to a near-ring group structure which motivates us to study various characteristics of the structure. If E is a quasi-injective N-group and S = End(injectivehull of E) then we study the structure ES and various properties of ES. It is proved that ES is a minimal quasi-injective exten- sion of E and any two minimal quasi-injective extensions are equivalent. This structure motivates to study the Jacobson radical of endomorphism near-ring of quasi-injective N-group E. It is established that the near-ring modulo the Jacobson radical is a regular near-ring. Some properties of quasi-injective N- groups relating essentially closed N-subgroups and complement N-subgroups are established.

Maximizing measures for endomorphisms of the circle

We study a given fixed continuous function and an endomorphism f : S1 → S1, whose f-invariant probability measures maximize . We prove that the set of endomorphisms having a ϕ maximizing invariant measure supported on a periodic orbit is dense.

Graphs of Morphisms of Graphs

This is an account for the combinatorially minded reader of various categories of directed and undirected graphs, and their analogies with the category of sets. As an application, the endomorphisms of a graph are in this context not only composable, giving a monoid structure, but also have a notion of adjacency, so that the set of endomorphisms is both a monoid and a graph. We extend Shrimpton's (unpublished) investigations on the morphism digraphs of reflexive digraphs to the undirected case by using an equivalence between a category of reflexive, undirected graphs and the category of reflexive, directed graphs with reversal. In so doing, we emphasise a picture of the elements of an undirected graph, as involving two types of edges with a single vertex, namely 'bands' and 'loops'. Such edges are distinguished by the behaviour of morphisms with respect to these elements.

A characterization of endotrivial modules over p -groups

Let P be a finite p-group, and let k a field of characteristic p>0. We prove that a finietly generated kG-module M is endotrivial if and only if where PHom (-,-) is the set of endomorphisms which factor through projective modules.

Herman rings in the theory of iterations of endomorphisms of C*

The article consists of two parts. In the first we investigate the problem of the existence of cycles of doubly-connected components of the Fatou set of endomorphisms of C *. By comparison with the cases of rational and entire functions, additional characteristics are found here. Section (c) of Theorem 1 and Example 1 were independently obtained by Baker [3] and by the author [4]. The basic result of the second part of the article is the theorem on J-instability of the endomorphisms of C * whose Fatou sets contain an invariant Herman ring (Theorem 2). For rational functions this result was proved by Mane in [5]. Mate's method of proof can be transferred to the case of C =* with almost no changes, but we shall prove Theorem 2 with the help of the method of "quasiconformal surgery" (this approach is possible also in the case of rational functions). This method first appeared in the works of Douady and Hubbard. Surgery by Herman rings was first applied by Shishikura in [6].

The geometry generated on homogeneous spaces by morphisms ofG-spaces

This survey is a continuation of the survey published in the Problems of Geometry Series, Vol. 15. It consists of an exposition of the general theory of homogeneous spaces defined by prescribing a set of endomorphisms of a Lie groupG. The morphisms of the homogeneous spaces so constructed are built up in the standard way and the geometry generated by these morphisms is studied.

Endomorphisms and null subalgebras of finite dimensional algebras

In [5] it was shown that a finite dimensional binary algebra without nilpotents over an algebraically closed field of characteristic zero has only the trivial derivation and, consequently, a finite automorphism group. It is shown here that an m-ary algebra without nilpotents has only finitely many endomorphisms, with no assumptions on the characteristic of the base field. An upper bound on the dimension of the algebraic set of endomorphisms of a finite dimensional m-ary algebra, depending on the dimension of a maximal null subalgebra, is obtained if the base field is the complex field. The null subalgebras arising from a single derivation are examined for the case of an algebraically closed base field of characteristic zero.

Complex Number Algebra as a Simple Case of Heaviside Operational Calculus

In [1] the author offered the following postulational basis for an algebraic structure called a heaviside operational calculus. Let G be an additive Abelian group of operands. Let E be a set of endomorphisms on G. If a nonnull element, U, of G has the property that every element of G is the map of U under some endomorphism in E, we call U a unit element of G relative to E. Suppose G and E satisfy the following postulates: 1. The endomnorphisms in E are permutable. 2. G contains at least one unit element, U, relative to E. 3. Any endomorphisin in E that sends a unit element U into a nonnull operand sends every nonnull operand in G into a nonnull operand. It follows, as shown in [1], that E is an integral domain under the usual definitions of equality, sums, and products for endomorphisms. For some choices of G and E, E will also be a field. G and E (or G and a class of operators associated with E) determine a heaviside operational calculus. For example, let G be the class of entering, sectionally continuous functions of t. The unit step function, {u(t) }, will serve as a unit operand for postulate 2. Given a member, f, of G we find the following endomorphism on G that sends u intof:

Examples of semirings of endomorphisms of semigroups

Semirings of endomorphisms of semigroups form an important class of semirings. All examples which can be found in the literature concern semigroups with a subcommutative operation. We show that there exists a non-subcommutative semigroup whose endomorphisms form a semiring (this answers a question raised by Professor A. H. Clifford). We also give an example of a semigroup whose set of endomorphisms is not embeddable into a semiring; however it is the disjoint union of two semirings, but one of these semirings is not embeddable into a semiring with identity.

A Note on Endomorphism Semigroups

If is a universal algebra, the set of endomorphisms of forms a monoid (i.e., semigroup with identity) under composition. We denote it by End (). For definitions and notations, see [1]. It is well known (e.g., [1], Theorem 12.3) that for any monoid M there is a unary algebra with M ≅ End (). E. Mendelsohn and Z. Hedrlin [3] have proved that the monoid of a subalgebra of an algebra is independent of the monoid of . In [2], Hedrlin proves the same for the monoid of a homomorphic image of . The proofs of these depend heavily on graph-theoretical and category-theoretical considerations. In this note considerably shorter direct algebraic proofs are given of these results.

VI.—Endomorphisms of Abstract Algebras

SynopsisA study is made of the generalization of the entropic law (xy)(zw)—(xz)(yw) to an identity connecting two operations. It is shown that such an identity is equivalent “in the large” to the condition that the set of endomorphisms with respect to one operation is closed with respect to the other. Furthermore, for such entropic operations, each may be regarded as a generalized endomorphism of the other and various generalizations of elementary properties of endomorphisms are obtained. Examples of entropic pairs of operations are quite common in mathematics and a number of these are discussed. An important aspect of the paper is the matrix notationintroduced to facilitate what would otherwise be extremely cumbersome computations with entropic operations.

A note on the system generated by a set of endomorphisms of a group.

A note on the system generated by a set of endomorphisms of a group.

A Representation for Boolean Algebras

1. The content. The success of Stone [5] in representing Boolean algebras by fields of sets leads one to search for other representations. (See also [2, p. 159], [4].) Using splitting endomorphisms on a type of Abelian group with an order relation, we are led to a faithful representation of a given Boolean algebra B. The group employed will be called a Boolean group. A special case of a Boolean group is a Boolean ring, and we shall show that 8 can be represented by a set of endomorphisms on its corresponding Boolean ring. If ? is complete the group on which it operates will be lower complete and upper conditionally complete. The Boolean groups are themselves special cases of a wider class of groups with an order relation, the vector ordered groups. Vector ordered groups generalize direct sums of Abelian groups, and their splitting endomorphisms form Boolean algebras. A simple example illustrates virtually the entire theory: Let ZJ be the ring of integers modulo 2, and let (M be the Cartesian product of some infinite collection of copies of 2*. 5 can be made into the strong direct sum of the 32 by introducing component-wise addition and multiplication. Then @ becomes a Boolean ring with a unity. It is also a Boolean algebra [2, p. 154] if one writes x ? y whenever each component of y ? (5 equals the corresponding component of x ? (M or is zero. Let 'P (() be the set of all endomorphisms V of the group structure of (M for which 12 _ v and V (x) ? x for every x P (M. If an order relation is defined in 'P in the obvious way, 'P becomes a Boolean algebra isomorphic to the Boolean algebra 5, so that the algebra 05 is repre