Partial Endomorphisms Research Articles

On partial endomorphisms of a star graph

In 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.

Partial automorphisms and injective partial endomorphisms of a finite undirected path

In this paper, we study partial automorphisms and, more generally, injective partial endomorphisms of a finite undirected path from Semigroup Theory perspective. Our main objective is to give formulas for the ranks of the monoids $$\mathrm {IEnd}(P_n)$$ and $$\mathrm {PAut}(P_n)$$ of all injective partial endomorphisms and of all partial automorphisms of the undirected path $$P_n$$ with n vertices. We also describe Green’s relations of $$\mathrm {PAut}(P_n)$$ and $$\mathrm {IEnd}(P_n)$$ and calculate their cardinals.

Modules whose partial endomorphisms have a δ-small kernels

Let R be a commutative ring and M a unital R-module. A submodule N is said to be δ-small, if whenever N + L = M with M/L is singular, we have L = M. M is called δ-small monoform if any of its partial endomorphism has δ-small kernel. In this paper, we introduce the concept of δ-small monoform modules as a generalization of monoform modules and give some of their properties, examples and characterizations.

Free idempotent generated semigroups: subsemigroups, retracts and maximal subgroups

ABSTRACTLet S be a subsemigroup of a semigroup T and let IG(E) and IG(F) be the free idempotent generated semigroups over the biordered sets of idempotents of E of S and F of T, respectively. We examine the relationship between IG(E) and IG(F), including the case where S is a retract of T. We give sufficient conditions satisfied by T and S such that for any e∈E, the maximal subgroup of IG(E) with identity e is isomorphic to the corresponding maximal subgroup of IG(F). We then apply this result to some special cases and, in particular, to that of the partial endomorphism monoid PEnd A and the endomorphism monoid EndA of an independence algebra A of finite rank. As a corollary, we obtain Dolinka’s reduction result for the case where A is a finite set.

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.

The Crossed Product by a Partial Endomorphism

Given a closed ideal I in a C*-algebra A, an ideal J (not necessarily closed) in I , a *-homomorphism α : A → M(I ) and a map L : J → A with some properties, based on earlier works of Pimsner and Katsura, we define a C*-algebra $$ \mathcal{O}{\left( {A,\alpha ,L} \right)} $$ which we call the Crossed Product by a Partial Endomorphism. We introduce the Crossed Product by a Partial Endomorphism $$ \mathcal{O}{\left( {X,\alpha ,L} \right)} $$ induced by a local homeomorphism σ : U → X where X is a compact Hausdorff space and U is an open subset of X. A bijection between the gauge invariant ideals of $$ \mathcal{O}{\left( {X,\alpha ,L} \right)} $$ and the σ, σ-1- invariant open subsets of X is showed. If (X, σ) has the property that $$ {\left( {{X}\ifmmode{'}\else$'$\fi,\sigma _{{\left| {{X}\ifmmode{'}\else$'$\fi} \right.}} } \right)} $$ is topologically free for each closed σ, σ-1-invariant subset X′ of X then we obtain a bijection between the ideals of $$ \mathcal{O}{\left( {X,\alpha ,L} \right)} $$ and the open σ, σ-1-invariant subsets of X.

The crossed product by a partial endomorphism and the covariance algebra

Given a local homeomorphism σ : U → X where U ⊆ X is clopen and X is a compact and Hausdorff topological space, we obtain the possible transfer operators L ρ which may occur for α : C ( X ) → C ( U ) given by α ( f ) = f ○ σ . We obtain examples of partial dynamical systems ( X A , σ A ) such that the construction of the covariance algebra C ∗ ( X A , σ A ) , proposed by B.K. Kwasniewski, and the crossed product by a partial endomorphism O ( X A , α , L ) , recently introduced by the author and R. Exel, associated to this system are not equivalent, in the sense that there does not exist an invertible function ρ ∈ C ( U ) such that O ( X A , α , L ρ ) ≅ C ∗ ( X A , σ A ) .

Dual categories for endodualisable Heyting algebras: optimization and axiomatization

This paper is both a contribution to natural duality theory as it applies to varieties of Heyting algebras and to the theory of standard topological quasi-varieties (see [3]) in general. We prove that the n-element Heyting chain C n has an alter ego, consisting of n−2 endomorphisms, that yields an optimal duality on the variety generated by C n . In the case n = 4, we give a set of quasi-equations that describe the dual category. We also give a set of quasi-equations that describe the strong dual category that is obtained by adding a partial endomorphism of C4 to the type of the alter ego. A quasi-equational description of the optimal dual category for the variety of Heyting algebras generated by \({\bf 2}^{2} \oplus {\bf 1}\) is also given. En route, we prove that if a finite topological unary algebra is injective amongst the finite algebras in the variety it generates, then it is standard. The results of [3] then allow us to give non-topological proofs of our topological descriptions of the dual categories.

Optimal natural dualities: the role of endomorphisms

AbstractThe optimality of dualities on a quasivariety , generated by a finite algebra , has been introduced by Davey and Priestley in the 1990s. Since every optimal duality is determined by a transversal of a certain family of subsets of Ω, where Ω is a given set of relations yielding a duality on , an understanding of the structures of these subsets—known as globally minimal failsets—was required. A complete description of globally minimal failsets which do not contain partial endomorphisms has recently been given by the author and H. A. Priestley. Here we are concerned with globally minimal failsets containing endomorphisms. We aim to explain what seems to be a pattern in the way endomorphisms belong to these failsets. This paper also gives a complete description of globally minimal failsets whose minimal elements are automorphisms, when is a subdirectly irreducible lattice-structured algebra.

OPTIMAL NATURAL DUALITIES: THE STRUCTURE OF FAILSETS

B. A. Davey and H. A. Priestley have investigated the optimality of dualities on a quasivariety [Formula: see text], where [Formula: see text] is a finite algebra. Relative to a given set Ω of relations yielding a duality, they characterized the optimal dualities as the dualities determined by the transversals of a certain family of subsets of Ω. However the structure of these subsets — known as globally minimal failsets — remained to be understood. This paper gives a complete description of the globally minimal failsets which do not contain partial endomorphisms, and an algorithmic method to determine them. These results are applied, by way of illustration, to the variety of de Morgan algebras and to two further varieties, one of them an Ockham algebra variety and the other a variety of Heyting algebras. All the globally minimal failsets are determined in each case.

On Definability of Hypergraphs by Their Semigroups of Homomorphisms

In studying mathematical objects on the basis of their mapping systems considerable attention is given to the endomorphism semigroups and the partial endomorphisms semigroups of the investigated objects. It is natural to apply the same attention to systems of partial homomorphisms of these objects. However, such systems were rarely considered, since in these situations the usual composition of binary relations cannot be used and it is necessary to consider such mapping systems together with other operations. Experience shows that it is possible to consider the operations of right restrictive multiplication and left restrictive multiplication introduced by Wagner in [14] in the setting of operations for systems of partial homomorphisms. For example, Schein [11] and Molchanov [8], [9] showed that in individual cases systems of partial homomorphisms endowed with the restrictive multiplications have simple structure and contain important information about the investigated objects. This paper continues investigations in this direction. We describe here a wide class of hypergraphs which are determined up to isomorphism by their restrictive semigroups of partial homomorphisms. We also apply this result in analysing some problems of endomorphism semigroups of hypergraphs. The main result of the paper — theorem 3.1 — was announced in [7]. Note also that results of this paper generalize and correct a series of statements formulated without proofs in the end of the survey article [10]. The author wishes to thank Professor Boris M. Schein and the referee of this paper for their numerous valuable suggestions.

Catalan monoids, monoids of local endomorphisms, and their presentations

The Catalan monoid and partial Catalan monoid of a directed graph are introduced. Also introduced is the notion of a local endomorphism of a tree, and it is shown that the Catalan (resp. partial Catalan) monoid of a tree is simply its monoid of extensive local endomorphisms (resp. partial endomorphisms) of finite shift. The main results of this paper are presentations for the Catalan and partial Catalan monoids of a tree. Our presentation for the Catalan monoid of a tree is used to give an alternative proof for a result of Higgins. We also identify results of Aizenstat and Popova which give presentations for the Catalan monoid and partial Catalan monoid of a finite symmetric chain.

Partial derivatives and endomorphisms of some relatively free Lie algebras

Partial derivatives and endomorphisms of some relatively free Lie algebras

On Extensions of Weakly Primitive Rings

If R is a ring an R-module M is called compressible when it can be embedded in each of its non-zero submodules; and M is called monoform if each partial endomorphism N → M, N ⊆ M, is either zero or monic. The ring R is called (left) weakly primitive if it has a faithful monoform compressible left module. It is known that a version of the Jacobson density theorem holds for weakly primitive rings [4], and that weak primitivity is a Mori ta invariant and is inherited by a variety of subrings and matrix rings. The purpose of this paper is to show that weak primitivity is preserved under formation of polynomials, rings of quotients, and group rings of torsion-free abelian groups. The key result is that R[x] is weakly primitive when R is (Theorem 1).

Ultimately periodic extension of partial endomorphisms of groups

Necessary and sufficient conditions for extending a partial endomorphism μ of a groupG to a total endomorphismμ * of a supergroup ofG are known. In this paper conditions, which are both necessary and sufficient, forμ * to be ultimately periodic are obtained.

Related representation theorems for rings, semi-rings, near-rings and semi-near-rings by partial transformations and partial endomorphisms

Fundamental statements for (associative) rings are that (a) the endomorphisms of each commutative group (U, +) form a ring and (b) eachring may be embedded in such a ring of endomorphisms. In order to generalise these theorems to groups and rings whose addition may not be commutative, one has to deal with partial endomorphisms. But thesering-theoretical Theorems 4a and 4b turn out to be specialisations of similarones for semi-near-rings, near-rings and semirings, developed here inSection 2 after some preliminaries on semi-near-rings in Section 1. A glance at Definition 1 and the ring-theoretical theorems and remarks at the end of Section 2 may give more orientation.

Semigroups of order-preserving partial endomorphisms on trees, I

Semigroups of order-preserving partial endomorphisms on trees, I

On Commutative Continuation of Partial Endomorphisms of Groups

Given a homomorphic mapping θ of a subgroup A of a group G onto another subgroup B of G, necessary and sufficient conditions for the existence of a supergroup G* of G and an endomorphism θ* of G* such that θ* coincides with θ on A were derived by B. H. Neumann and Hanna Neumann (3). The homomorphism θ is called a partial endomorphism of G and θ* is said to continue, or extend, θ. Necessary and sufficient conditions for the simultaneous continuation of two partial endomorphisms of a group G to total endomorphisms of one supergroup G* ⊇ G were derived by the author (2).

Extension of partial endomorphisms of abelian groups

It is known [1] that for a partial endomorphism μ of a group G that maps the subgroup A ⊆ G onto B ⊆ G. G to be extendable to a total endomorphism μ* of a supergroup G* ⊆ G such that μ an isomorphism on G*(μ*)m for some positive integer m, it is necessary and sufficient that there exist in G a sequence of normal subgroupssuch that L1 ƞA is the kernel of μ andfor ι = 1, 2,…, m–1.The question then arises whether these conditions could be simplified when the group G is abelian. In this paper it is shown not only that the conditions are simplified when Gis abelian but also that the extension group G*⊇G can be chosen as an abelian group.

Embedding Theorems for Abelian Groups

Given an abelian group G and a mapping θ that maps a subgroup A of G homomorphically onto another subgroup B of G, then it is known (3) that there always exists an embedding group G* ⊇ G which is abelian and possesses an endomorphism θ* which coincides with θ on A, i.e. aθ = aθ* whenever aθ is defined. θ is called a partial endomorphism of G and θη a total endomorphism or simply an endomorphism that extends or continues θ.