Semigroup Of Endomorphisms Research Articles

The automorphisms of endomorphism semigroups of free burnside groups

In this paper, we describe the automorphism groups of the endomorphism semigroups of free Burnside groups B(m, n) for odd exponents n ≥ 1003. We prove, that the groups Aut (B(m, n)) and Aut ( End (B(m, n))) are canonically isomorphic. In particular, if the groups Aut ( End (B(m, n))) and Aut ( End (B(k, n))) are isomorphic, then m = k.

Continuous unital dilations of completely positive semigroups

Since it began in the 1970s, the study of completely positive semigroups has included among its central topics the dilation of a completely positive semigroup to an endomorphism semigroup. Several authors have proved the existence of dilations, but have usually only achieved unital dilations of specific examples, while general theorems have tended to produce non-unital dilations. A unique approach due to Jean-Luc Sauvageot leads to a unital dilation in a general setting, but leaves unclear the continuity of the dilation semigroup. The major purpose of this paper, therefore, is to further develop Sauvageot's theory in order to prove the existence of continuous unital dilations. The dilation is characterized by a combinatorial property akin to free independence. This property is implicit in some of Sauvageot's original calculations, but a secondary goal of this paper is to present it as its own object of study, which is done in Section 4 . This work is based on the author's Ph.D. thesis.

Exotic one-parameter semigroups of endomorphisms of a symmetric cone

We construct an exotic one-parameter semigroup of endomorphisms of a symmetric cone C, whose generator is not the sum of a Lie group generator and an endomorphism of C. The question is motivated by the theory of affine processes on symmetric cones, which plays an important role in mathematical finance. On the other hand, theoretical question that we solve in this paper seems to have been implicitly open even much longer then this motivation suggests.

Mixed Abelian groups with isomorphic endomorphism semigroups

In the present paper, mixed Abelian groups with isomorphic endomorphism semigroups are studied. In particular, we characterize the groups whose periodic parts are nonisomorphic, while their endomorphism semigroups are isomorphic. A description of nonreduced split mixed Abelian groups with UA-rings of endomorphisms is obtained.

Automorphism Groups and Endomorphism Semigroups of Groups B(m, n)

Automorphism Groups and Endomorphism Semigroups of Groups B(m, n)

Correspondences of the semigroup of endomorphisms of an equivalence relation

In terms of different constructions of the wreath product of a transformation semigroup with a small category and of the direct product of wreath products of groups, faithful representations of three correspondences of the semigroup of all endomorphisms of an arbitrary equivalence relation are described, namely, of the semigroup of all endotopisms, of the monoid of all strong endotopisms, and, correspondingly, of the group of all autotopisms of a given equivalence relation.

STOPPING THE CCR FLOW AND ITS ISOMETRIC COCYCLES

It is shown how to use non-commutative stopping times in order to stop the CCR flow of arbitrary index and also its isometric cocycles, i.e. left operator Markovian cocycles on Boson Fock space. Stopping the CCR flow yields a homomorphism from the semigroup of stopping times, equipped with the convolution product, into the semigroup of unital endomorphisms of the von Neumann algebra of bounded operators on the ambient Fock space. The operators produced by stopping cocycles themselves satisfy a cocycle relation.

Primitive groups synchronize non-uniform maps of extreme ranks

Let Ω be a set of cardinality n, G a permutation group on Ω, and f:Ω→Ω a map which is not a permutation. We say that G synchronizes f if the semigroup 〈G,f〉 contains a constant map.The first author has conjectured that a primitive group synchronizes any map whose kernel is non-uniform. Rystsov proved one instance of this conjecture, namely, degree n primitive groups synchronize maps of rank n−1 (thus, maps with kernel type (2,1,…,1)). We prove some extensions of Rystsov's result, including this: a primitive group synchronizes every map whose kernel type is (k,1,…,1). Incidentally this result provides a new characterization of imprimitive groups. We also prove that the conjecture above holds for maps of extreme ranks, that is, ranks 3, 4 and n−2.These proofs use a graph-theoretic technique due to the second author: a transformation semigroup fails to contain a constant map if and only if it is contained in the endomorphism semigroup of a non-null (simple undirected) graph.The paper finishes with a number of open problems, whose solutions will certainly require very delicate graph theoretical considerations.

On endomorphisms of groups of order 32 with maximal subgroups C4 x C2 x C2

It is proved that each group of order 32 which has a maximal subgroup isomorphic to C8 £ C2 is determined by its endomorphism semigroup in the class of all groups.

Riemann surfaces in Stein manifolds with the Density property

We show that any open Riemann surface can be properly immersed in any Stein manifold with the (Volume) Density property and of dimension at least 2. If the dimension is at least 3, we can actually choose this immersion to be an embedding. As an application, we show that Stein manifolds with the (Volume) Density property and of dimension at least 3, are characterized among all other complex manifolds by their semigroup of holomorphic endomorphisms.

Varieties characterized by their endomorphisms

We show that two varietes X and Y with isomorphic endomorphism semigroups are isomorphic up to field automorphism if one of them is affine and contains a copy of the affine line. A holomorphic version of this result is due to the first author.

Semigroups of Strong Endomorphisms of Infinite Graphs and Hypergraphs

We define a class of infinite undirected graphs and a class of infinite n-regular hypergraphs and prove that any semigroup of all strong endomorphisms of the graphs and hypergraphs from these classes is isomorphic to the wreath product of a transformation monoid and a small category. We establish the criterional conditions for the regularity of the semigroup of strong endomorphisms of infinite n-regular hypergraphs.

Semigroups of endomorphisms of some infinite monounary algebras

We consider a certain class of infinite monounary algebras that do not contain cyclic and metamonogenic subalgebras. We show that a monounary algebra of this class is determined up to isomorphism by its semigroup of endomorphisms. It is also proved that, in the semigroup of endomorphisms of every monounary algebra of this class, there exists a densely imbedded ideal, whose cardinal number is equal to the order of this monounary algebra, and which also determines it.

Endomorphism semigroups of some free products

We define the notion of a w-class of semigroups and prove that every endomorphism semigroup of a free product of semigroups from a maximal w-class is isomorphic to a wreath product of a transformation semigroup with some small category.

The endomorphism semigroup of an endomapping of a finite set

The structure of the semigroup of all endomorphisms of an endomapping of a finite set has been determined. This has been done by naturally representing the endomapping by a directed graph, and determining the structure of the endomorphism semigroup of this graph.DOI: http://dx.doi.org/10.3329/ganit.v31i0.10310GANIT J. Bangladesh Math. Soc. (ISSN 1606-3694) 31 (2011) 71-77

Definition of positively closed classes by endomorphism semigroups

The positively closed classes of multivalued logic are characterised with the use of endomorphism semigroups. All subsemigroups of the symmetric semigroup of degree k are described which are semigroups of endomorphisms of positively closed classes in P k . On the base of the obtained results, we find all positively precomplete classes in P k .

Automorphisms of partial endomorphism semigroups

In this paper we propose a general recipe for calculating the automorphism groups of semigroups consisting of partial endomorphisms of relational structures over a finite set with a single m-ary relation for any m E N. We use this recipe to determine the automorphism groups of the following semigroups: the full transformation semigroup, the partial transformation semigroup, and the symmetric inverse semigroup, the wreath product of two full transformation semigroups, the partial endomorphisms of any partially ordered set, the full spectrum of semigroups of partial mappings preserving or reversing a linear or circular order.

Automorphisms of the endomorphism semigroup of a polynomial algebra

We describe the automorphism group of the endomorphism semigroup End ( K [ x 1 , … , x n ] ) of ring K [ x 1 , … , x n ] of polynomials over an arbitrary field K. A similar result is obtained for automorphism group of the category of finitely generated free commutative–associative algebras of the variety CA commutative algebras. This solves two problems posed by B. Plotkin (2003) [18, Problems 12 and 15]. More precisely, we prove that if φ ∈ Aut End ( K [ x 1 , … , x n ] ) then there exists a semi-linear automorphism s : K [ x 1 , … , x n ] → K [ x 1 , … , x n ] such that φ ( g ) = s ∘ g ∘ s − 1 for any g ∈ End ( K [ x 1 , … , x n ] ) . This extends the result obtained by A. Berzins for an infinite field K.

On determinability of idempotent medial commutative quasigroups by their endomorphism semigroups

We extend the result of P. Puusemp (Idempotents of the endomorphism semigroups of groups. Acta Comment. Univ. Tartuensis, 1975, 366, 76-104) about determinability of finite Abelian groups by their endomorphism semigroups to finite idempotent medial commutative quasigroups.

On the semigroups of endomorphisms of direct products of commutative Moufang loops

In this paper it is proved that the semigroup of endomorphisms of a commutative Moufang loop (CML) which admits a decomposition into the direct product of its subloops is isomorphic to the semigroup of M, M-matrices. We introduce the notion of the holomorph of a CML which admits a direct decomposition and give the matrix representation of the holomorph.