Finite Orbits of Language Operations

We consider a set of natural operations on languages, and prove that the orbit of any language L under the monoid generated by this set is finite and bounded, independently of L. This generalizes previous results about complement, Kleene closure, and positive closure

A Linear Version of Kuratowski Problem

  El presente estudio se enmarca dentro del paradigma de la investigación básica matemática. Nuestro trabajo está orientado hacia el planteamiento de una nueva versión conceptual y la construcción de una prueba formal del reconocido problema de clausura-complemento de Kuratowski en el marco de la teoría de los espacios vectoriales.   Palabras clave: Espacios vectoriales, Problema de Kuratowski, problema de cerradura y complemento.   Abstract This study is framed into the basic mathematics research paradigm. This work is geared towards the statement of a new conceptual version and to the construction of a formal proof of the well-known Kuratowski closure-complement problem in the vector space theory frame.   Keywords: Kuratowski’s problem, Vector spaces, Closure-complement problem   Resumo O presente estudo está enquadrado no paradigma da pesquisa matemática básica. Este trabalho está voltado para afirmação de uma nova versão conceitual e para a construção de uma prova formal do bem conhecido problema de complemento de roupas de Kuratowski no quadro da teoria linear do espaço.   Palavras chave: Problema de Kuratowski, Espaços vetoriais, Problema de fechamento-complemento &nbsp

A Term Rewriting System for Kuratowski\u27s Closure-Complement Problem

We present a term rewriting system to solve a class of open problems that are generalisations of Kuratowski\u27s closure-complement theorem. The problems are concerned with finding the number of distinct sets that can be obtained by applying combinations of axiomatically defined set operators. While the original problem considers only closure and complement of a topological space as operators, it can be generalised by adding operators and varying axiomatisation. We model these axioms as rewrite rules and construct a rewriting system that allows us to close some so far open variants of Kuratowski\u27s problem by analysing several million inference steps on a typical personal computer

The monoid consisting of Kuratowski operations

The paper fills gaps in knowledge about Kuratowski operations which are already in the literature. The Cayley table for these operations has been drawn up. Techniques, using only paper and pencil, to point out all semigroups and its isomorphic types are applied. Some results apply only to topology, one can not bring them out, using only properties of the complement and a closure-like operation. The arguments are by systematic study of possibilities.Comment: We are going to submit the article to a journa

Boundary-Border Extensions of the Kuratowski Monoid

The Kuratowski monoid $\mathbf{K}$ is generated under operator composition by closure and complement in a nonempty topological space. It satisfies $2\leq|\mathbf{K}|\leq14$. The Gaida-Eremenko (or GE) monoid $\mathbf{KF}$ extends $\mathbf{K}$ by adding the boundary operator. It satisfies $4\leq|\mathbf{KF}|\leq34$. We show that when $|\mathbf{K}|<14$ the GE monoid is determined by $\mathbf{K}$. When $|\mathbf{K}|=14$ if the interior of the boundary of every subset is clopen, then $|\mathbf{KF}|=28$. This defines a new type of topological space we call $Kuratowski\ disconnected$. Otherwise $|\mathbf{KF}|=34$. When applied to an arbitrary subset the GE monoid collapses in one of $70$ possible ways. We investigate how these collapses and $\mathbf{KF}$ interdepend, settling two questions raised by Gardner and Jackson. Computer experimentation played a key role in our research.Comment: 48 pages, 9 figure

Zastosowania teorii kategorii w topologii ogólnej

This dissertation is concerned with some notions of category theory in application to general topology and investigates to what extent the arguments making use of those notions can replace a more traditional approach. One of the generic examples is the notion of Fraisse sequence. Quite a number of categories consist of algebraic objects: groups, Abelian groups and monoids with group, Abelian group and monoid homomorphisms, respectively. An example of a monoid consisting of topological operations is the Kuratowski monoid. The dissertation is organized as follows. In subsection 1.2.1 we give some comments on ways of making use of concepts such as a tree, the space of branches, an ultra-metric and so forth. In section 2 we provide axioms for categories, basic definitions, notations and examples. We draw attention to a particular covariant functor, namely a diagram. At the beginning of section 3 we remind basic ideas concerning the limit of inverse sequence. Then we analyze a proof of Knaster-Reichbach theorem about extendibility of a mapping between two closed, nowhere dense subsets of the Cantor set and its version for scattered compact metric spaces. In section 4 we focus on a symmetric Cantorval. Beside the Cantor set and the sum of a finite family of closed intervals it is one of the possible forms the set of subsums of a convergent series with positive terms can take. In section 5 we remind the notions of the amalgamation and the inverse amalgamation property as well as their special cases - pushout and pullback, respectively. We discuss 1 these notions in two particular categories: S e t category whose objects are sets and whose arrows are functions and T op category whose objects are topological spaces and whose arrows are continuous mappings. In section 6 we present the main idea of the dissertation, namely Fraisse sequences: definition, criterion of existence and some properties. In subsection 6.2 we focus on the category F in s e top/A, i.e. a comma-category related to the opposite category of the category F in s e t . In section 7 we discuss the structure and basic facts concerning Fraisse limits in the context of inverse sequences. We show a way in which the properties of projective universality and projective homogeneity combine with the concept of generic object (for a given subcategory). Section 8 comprises comments th a t will be included in joint papers with the supervisors concerning, among others, a category of finite non-empty sets and its linear version. We claim here th a t the Cantor set admits a unique strictly positive probability measure th a t takes rationa values on clopen sets and satisfies certain homogeneity condition