Class Of Morphisms Research Articles

Representation of rational functions with prefix and suffix codings

We proceed with the characterization of rational functions by means of restricted class of morphisms. Left subsequential transductions can be factored in an endmarking followed by an uniform morphism, the inverse of a prefix morphism and an alphabetic morphism. Rational functions require the inverse of a prefix morphism followed by the inverse of a suffix morphism.

An algebraic characterisation of elementary net system (observable) state space

Abstract The paper proposes a characterisation of the state space ofan Elementary Net (EN) system in terms of a relational algebraic structure, called Local State Transformation (LST) algebra. The latter allows one to take into consideration in an explicit way local state transformations performed by the occurrence of events. A class of injective morphisms between LST algebras allows one to compare system behaviours at different levels of abstraction and to introduce a notion of preorder in the class of EN systems. This leads in a natural way to a notion of equivalence which preserves the system degree of concurrency, and whose equivalence classes have a unique (up to isomorphism) canonical representative which is minimal with respect to net elements. Moreover, the class of morphisms allows the definition of a notion of state observability and of a notion of equivalence between Observable Elementary Net Systems. All these notions provide the system designer with a set of useful tools for an incremental design process, in which the (concurrent and nondeterministic) control of the considered system need not be fixed ‘a priori’.

Factorization, diagonal separation, and disconnectedness

Let ( E, M) be a factorization structure on C , a category with finite products. C M is the full subcategory of C such that X ϵ C M if and only if δ x ϵ M. We describe a class of morphisms M A such that ( A -epi, M A ) is a factorization structure on C if C also has intersections and equalizers, and show that A = C M for some factorization structure on C if and only if A = C M A . We then look at relative disconnectednesses and disconnectednesses in extremally well-powered and strongly complete categories, and show their relationship to the ( A -epi, M A ) factorization structure.

A factorization of the Pumplün-Röhrl connection

The Galois connection given in 1985 by Pumplün and Röhrl between the classes of objects and the classes of morphisms in any category is shown (under ordinary circumstances) to have a “natural” factorization through the system of all idempotent closure operators over the category. Futhermore, each “component” of the factorization is a Galois connection in its own right. The first factor is obtained by using a generalization of the process, given by Salbany in 1975, that yields a closure operator for any class of topological spaces, while the second factor can be used to form the weakly hereditary core of an idempotent closure operator.

Fibonacci morphisms and Sturmian words

Márton Kósa (1987) has stated five conjectures devoted to the study of some binary morphisms and the factors of infinite words these morphisms generate. Positive answers to the first four of these conjectures are given in this paper, the main result being the characterization of a class of morphisms whose stationary words are Sturmian.

Decomposition techniques for finite semigroups, using categories II

In this paper we study in a systematic fashion the decomposition of semigroup morphisms θ: S→ T with S and T finite, in the form θ= ϕπ, where ϕ: S< V n ∫( V n − 1 ∫⋯( V 1∫ T)⋯). Here, the V i 's are monoids, ∫ denotes either the semidirect or the 2-sided semidirect product, and π denotes the canonical projection of V n ∫( V n − 1 ∫⋯( V 1∫ T)⋯) onto T. These results are then refined for special classes of morphisms, and in particular for aperiodic morphisms and LG-morphisms.

Factorizations, denseness, separation, and relatively compact objects

Some of the relationships among the topological notions: ‘Hausdorff’, ‘compact’, ‘perfect’, and ‘closed’ are abstracted to a more general categorical setting, where they are shown to remain intact. An investigation is made of factorization structures (especially for single morphisms) and their relationships to strong limit operators and to the Pumplu¨n-Ro¨hrl Galois correspondence between classes of objects and classes of morphisms in any category. Many examples as well as internal characterizations of Galois-closed classes are provided.

On left-cancellable classes of morphisms

On left-cancellable classes of morphisms

On weak monomorphisms in the category of convex processes

Convex processes have been introduced and studied by R.T. Rockafellar [3–5]. Some characterizations of monomorphisms and epimorphisms of the category of polyhedral convex processes have been given [6]. In the present paper we give a new class of morphisms in the category of all convex processes and we study the connections between this class and the class of monomorphisms.

Präradikale und Morphismenklassen

A preradical is a normal subfunctor of the identity-functor. In this note I give a correspondence between preradicals and classes of morphisms in rather general categories. For a classI of morphisms R I , defined by R I C=∩Ke (if|ieI and feMor (C, Range (i)) is a preradical and-all preradicals are of this type. As an interesting application I give a characterisation of the set of all left exact preradicals for A-modules. Also I have obtained results about quasi-injective A-modules for a Noetherian ring A. Ich danke Herrn Prof. Dr. Kasch fur seine Anregungen, den Gutachtern fur ihre Hinweise.

Rigidity of the $K(1)$-local stable homotopy category

One goal of algebraic topology is to find algebraic invariants that classify topological spaces up to homotopy equivalence. The notion of homotopy is not only restricted to topology. It also appears in algebra, for example as a chain homotopy between two maps of chain complexes. The theory of model categories,introduced by D. Quillen [Qui06], provided us with a powerful common language to represent different notions of homotopy. Quillen’s work transformed algebraic topology from the study of topological spaces into a wider setting useful in many areas of mathematics, such as homological algebra and algebraic geometry, where homotopy theoretic approaches led to interesting results. In brief, a model structure on a category C is a choice of three distinguished classes of morphisms: weak equivalences, fibrations, and cofibrations, satisfying certain axioms. We can pass to the homotopy category Ho(C) associated to a model category C by inverting the weak equivalences, i.e. by making them into isomorphisms. While the axioms allow us to define the homotopy relations between classes of morphisms in C, the classes of fibrations and cofibrations provide us with a solution to the set-theoretic issues arising in general localisations of categories. Even though it is sometimes sufficient to work in the homotopy category, looking at the homotopy level alone does not provide us with enough higher order structure information. For example, homotopy (co)limits are not usually a homotopy invariant, and in order to define them we need the tools provided by the model category. This is where the question of rigidity may be asked: if we just had the structure of the homotopy category, how much of the underlying model structure can we recover? This question of rigidity has been investigated during the last decade, and an extremely small list of examples have been studied, which leaves us with a lot of open questions regarding this fascinating subject. Ou goal is to investigate one of the open questions which have not been answered before. In this thesis, we prove rigidity of the K(1)-local stable homotopy category Ho(LK(1)Sp) at p = 2. In other words, we show that recovering higher order structure information, which is meant to be lost on the homotopy level,is possible by just looking at the triangulated structure of Ho(LK(1)Sp). This new result does not only add one more example to the list of known examples of rigidity in stable homotopy theory, it is also the first studied case of rigidity in the world of Morava K-theory.