Dikran Dikranjan

Closure operators I

Abstract Closure operators in an ( E , M )-category X are introduced as concrete endofunctors of the comma category whose objects are the elements of M . Various kinds of closure operators are studied. There is a Galois equivalence between the conglomerate of idempotent and weakly hereditary closure operators of X and the conglomerate of subclasses of M which are part of a factorization system. There is a one-to-one correspondence between the class of regular closure operators and the class of strongly epireflective subcategories of X . Every closure operators admits an idempotent hull and a weakly hereditary core. Various examples of additive closure operators in Top are given. For abelian categories standard closure operators are considered. It is shown that there is a one-to-one correspondence between the class of standard closure operators and the class of preradicals. Idempotent, weakly hereditary, standard closure operators correspond to idempotent radicals (= torsion theories).

ALGEBRAIC ENTROPY FOR ABELIAN GROUPS

The theory of endomorphism rings of algebraic structures allows, in a natural way, a systematic approach based on the notion of entropy borrowed from dynamical systems. Here we study the algebraic entropy of the endomorphisms of Abelian groups, introduced in 1965 by Adler, Konheim and McAndrew. The so-called Addition Theorem is proved; this expresses the algebraic entropy of an endomorphism φ of a torsion group as the sum of the algebraic entropies of the restriction to a φ-invariant subgroup and of the endomorphism induced on the quotient group. Particular attention is paid to endomorphisms with zero algebraic entropy as well as to groups all of whose endomorphisms have zero algebraic entropy. The significance of this class arises from the fact that any group not in this class can be shown to have endomorphisms of infinite algebraic entropy, and we also investigate such groups. A uniqueness theorem for the algebraic entropy of endomorphisms of torsion Abelian groups is proved.

Algebraic structure of pseudocompact groups

Introduction Principal results Preliminaries Some algebraic and set-theoretic properties of pseudocompact groups Three technical lemmas Pseudocompact group topologies on

TOPOLOGICAL CATEGORIES AND CLOSURE OPERATORS

\mathcal V

Zero-dimensionality of some pseudocompact groups

-free groups Pseudocompact topologies on torsion Abelian groups Pseudocompact connected group topologies on Abelian groups Pseudocompact topologizations versus compact ones Some diagrams and open questions Diagram 2 Diagram 3 Bibliography.

Recent advances in minimal topological groups

Abstract It is shown that the category CS of closure spaces is a topological category. For each epireflective subcategory A of a topological category X a functor F A :X → X is defined and used to extend to the general case of topological categories some results given in [4], [5] and [10] for epireflective subcategories of the category Top of topological spaces.

Categorically compact topological groups

We prove that hereditarily disconnected countably compact groups are zero-dimensional. This gives a strongly positive answer to a question of Shakhmatov. We show that hereditary or total disconnectedness yields zerodimensionality in various classes of pseudocompact groups.

Chapter 41 - Selected topics from the structure theory of topological groups

Abstract This survey presents some recent trends and results (most of them unpublished) in minimal groups. The following are the main direction: 1. (a) permanence properties of minimal groups; 2. (b) complete minimal groups; 3. (c) countably compact minimal groups; 4. (d) algebraic structure of minimal abelian groups. In (a) we discuss preservation of minimality under the main group-theoretic operations: taking (direct or semidirect) products, quotients and (dense or closed) subgroups. Particular emphasis is given to infinite products and the critical power of minimality of a minimal abelian group G (this is the least nonminimal power of G provided such powers exist, otherwise κ ( G ) = 1). In particular, there exist (strongly) pseudocompact minimal abelian groups G with κ ( G ) = ω 1 , while for countably compact minimal abelian groups G either κ ( G ) = 1, when the connected component of G is compact, or κ ( G ) = ω ; otherwise. (b) and (c) are parallel but in opposite direction. In (b) we put completeness-like conditions on the minimal groups and see when they are compact. In (c) we impose countable compactness on the minimal groups and look for further conditions which may yield compactness. In this way (b) becomes a chase for precompactness , while (c) becomes a chase for completeness . This is why we dedicate in (b) special attention to the celebrated precompactness theorem for minimal abelian groups of Prodanov and Stoyanov and we offer some examples and comments in the case of nilpotent groups. Here we consider also stronger completeness conditions, as local compactness, completeness of all quotients, etc. It turns out that the question whether connected, countably compact, minimal abelian groups are compact depends on the existence of measurable cardinals. More precisely, the connected component of a countably compact minimal abelian group G must be compact whenever its size is not Ulam-measurable. In such a case κ ( G ) = 1 and the algebraic structure of G can be completely described. Under the assumption that there exist measurable cardinals one can construct a noncompact ω-bounded, connected, minimal abelian group G (this entails, of course, κ ( G ) = ω ). In (d) we give an alternative exposition of the known results on this question. Our approach takes into account the connection between the algebraic invariants of the group and its topological properties. We pay special attention to the case of abelian groups of free-rank c resolved by Schinkel (1990, Dissertation) and answer an open question of his regarding the case of torsion-free groups of large free-ranks. We show that ZFC cannot answer the question whether the free abelian group F of rank c admits minimal pseudocompact group topologies, even if F admits both (totally) minimal group topologies and pseudocompact group topologies.

The Markov–Zariski topology of an abelian group

Abstract We study the notion of a categorically compact topological group, suggested by the Kuratowski-Mrowka characterization of compact spaces. A topological group G is categorically compact, or C-compact, if for any topological group H the projection G × H → H sends closed subgroups to closed subgroups. We prove, among others, the following theorems: (1) any product of C-compact topological groups is C-compact; (2) separable C-compact groups are totally minimal; (3) C-compact soluble topological groups are compact.

Adjoint algebraic entropy

Publisher Summary This chapter discusses selected topics from the structure theory of topological groups. It contains open problems and questions covering the a number of topics including: the dimension theory of topological groups, pseudocompact and countably compact group topologies on Abelian groups, with or without nontrivial convergent sequences, categorically compact groups, sequentially complete groups, the Markov–Zariski topology, the Bohr topology, and transversal group topologies. All topological groups considered in this chapter are assumed to be Hausdorff. It is stated that Abelian group G is algebraically compact provided that an Abelian group H is found such that G × H admits a compact group topology. Algebraically compact groups form a relatively narrow subclass of Abelian groups (for example, the group ℤ of integers is not algebraically compact). On the other hand, every Abelian group G is algebraically pseudo-compact; that is, an Abelian group H can be found such that G × H ∈ P . Problems and questions related to Bohr-homeomorphic bounded Abelian groups are also discussed in the chapter.