Retract rational fields and cyclic Galois extensions

In [23], this author began a study of so-called lifting and approximation problems for Galois extensions. One primary point was the connection between these problems and Noether’s problem. In [24], a similar sort of study was begun for central simple algebras, with a connection to the center of generic matrices. In [25], the notion of retract rational field extension was defined, and a connection with lifting questions was claimed, which was used to complete the results in [23] and [24] about Noether's problem and generic matrices. In this paper we, first of all, set up a language which can be used to discuss lifting problems for very general “linear structures”. Retract rational extensions are defined, and proofs of their basic properties are supplied, including their connection with lifting. We also determine when the function fields of algebraic tori are retract rational, and use this to further study Noether’s problem and cyclic 2-power Galois extensions. Finally, we use the connection with lifting to show that ifp is a prime, then the center of thep degree generic division algebra is retract rational over the ground field. The author is grateful for NSF support under grant #MCS79-04473.     

On Hopf Galois Extension of Separable Algebras

In this paper, the classical Galois theory to the H*-Galois case is developed. Let H be a semisimple and cosemisimple Hopf algebra over a field k, A a left H-module algebra,and A/AHa right H*-Galois extension. The authors prove that, if AHis a separable kalgebra, then for any right coideal subalgebra B of H, the B-invariants A~B= {a ∈ A |b · a = ε(b)a, b∈ B} is a separable k-algebra. They also establish a Galois connection between right coideal subalgebras of H and separable subalgebras of A containing AHas in the classical case. The results are applied to the case H =(kG)*for a finite group G to get a Galois 1-1 correspondence.     

Galois theory for sets of operations closed under permutation,cylindrification, and composition

A set of operations on A is shown to be the set of linear term operations of some algebra on A if and only if it is closed under permutation of variables, addition of inessential variables, and composition, and if it contains all projections. A Galois framework is introduced to describe the sets of operations that are closed under the operations mentioned above, not necessarily containing all projections. The dual objects of this Galois connection are systems of pointed multisets, and the Galois closed sets of dual objects are described accordingly. Moreover, the closure systems associated with this Galois connection are shown to be uncountable (even if the closed sets of operations are assumed to contain all projections).     

Interior operators and topological separation

A notion of separation with respect to an interior operator in topology is introduced and some basic properties are presented. In particular, it is shown that this notion of separation with respect to an interior operator gives rise to a Galois connection between the collection of all subclasses of the class of topological spaces and the collection of all interior operators in topology. Characterizations of the fixed points of this Galois connection are given and examples are provided.     

A general Galois theory for cofunctions and corelations

Cofunctions and corelations are introduced and a corresponding Galois connection is studied in analogy to operations and relations in universal algebra. The Galois closed sets are characterized by local closures of clones of cofunctions and corelations, respectively. As an application concrete characterization problems are considered e.g. for strong bisimulations.     

A Link Between Two Connectedness Notions

The composition of two previously introduced Galois connections is used to provide a wider perspective on the Clementino–Tholen connectedness versus separation Galois connection. Moreover, a link between this and Castellini's connectedness–disconnectednes Galois connection is also presented.     

CONNECTEDNESS,DISCONNECTEDNESS AND CLOSURE OPERATORS: FURTHER RESULTS

Abstract

Let X be an arbitrary category with an (E, M)-factorization structure for sinks. A notion of constant morphism that depends on a chosen class of monomorphisms was previously used to provide a generalization of the connectedness-disconnectedness Galois connection (also called torsion-torsion free in algebraic contexts). This Galois connection was shown to factor through the class of all closure operators on X with respect to M. Here, properties and implications of this factorization are investigated. In particular, it is shown that this factorization can be further factored. Examples are provided.     

代数构造及在Ramsey理论中的应用

本文在Galois域上的代数构造和关于一些特定类型图的Ramseyr数之间建立了一个关系．关键问题是研究了关于Galois域上的代数构造的方程及方程组的解．我们得到了一些关于二部图的新的下界和上界．     

Galois self-dual extended duadic constacyclic codes

Galois inner product is a generalization of the Euclidean inner product and Hermitian inner product. The theory on linear codes under Galois inner product can be applied in the constructions of MDS codes and quantum error-correcting codes. In this paper, we construct Galois self-dual codes and MDS Galois self-dual codes from extensions of constacyclic codes. First, we explicitly determine all the Type II splittings leading to all the Type II duadic constacyclic codes in two cases. Second, we propose methods to extend two classes of constacyclic codes to obtain Galois self-dual codes, and we also provide existence conditions of Galois self-dual codes which are extensions of constacyclic codes. Finally, we construct some (almost) MDS Galois self-dual codes using the above results. Some Galois self-dual codes and (almost) MDS Galois self-dual codes obtained in this paper turn out to be new.     

A Boolean matrix iteration in timetable construction

The annoying experience in timetable construction is that usually a complete timetable cannot be found without violating or diminishing some preconditions, even if the problem is theoretically solvable. Neither the control of the Hall conditions by Gotlieb's process of reducing availabilities nor the application of elaborate exchange operations guarantees a solution. In this paper an iteration of elementary implications is described which is expected to improve this situation, if applied in the final period of construction. In the course of these investigations, some formulas on Boolean matrices are derived, and a Galois connection between sets of Boolean vectors and Boolean matrices is exhibited.     

CONNECTEDNESS AND DISCONNECTEDNESS: A DIFFERENT PERSPECTIVE

Abstract

A factorization of a Galois connection investigated earlier is used to give a definition of a connectedness-disconnectedness Galois connection that is free of the notion of constant morphism. A new notion of N-fixed morphism with respect to a class N of monomorphisms is presented. This is used to characterize the connectedness-disconnectedness Galois connection in the case that N is closed under the formation of pullbacks. Some closedness properties of these Galois connections are investigated.     

代数构造及在Ramsey理论中的应用

这篇文章在伽罗瓦域上的代数构造和关于一些特定类型图的Ramsey数之间建立了一个关系. 研究了关于伽罗瓦域上的代数构造的方程及方程组的解. 我们得到了一些关于二部图的Ramsey数的新的下界和上界.     

Fundamental groupoids for simplicial objects in Mal'tsev categories

We show that the category of internal groupoids in an exact Mal'tsev category is reflective, and, moreover, a Birkhoff subcategory of the category of simplicial objects. We then characterize the central extensions of the corresponding Galois structure, and show that regular epimorphisms admit a relative monotone-light factorization system in the sense of Chikhladze. We also draw some comparison with Kan complexes. By comparing the reflections of simplicial objects and reflexive graphs into groupoids, we exhibit a connection with weighted commutators (as defined by Gran, Janelidze and Ursini).     

Fuzzy Galois Connections

The concept of Galois connection between power sets is generalized from the point of view of fuzzy logic. Studied is the case where the structure of truth values forms a complete residuated lattice. It is proved that fuzzy Galois connections are in one-to-one correspondence with binary fuzzy relations. A representation of fuzzy Galois connections by (classical) Galois connections is provided.     

基于格值逻辑的模糊概念格

从逻辑的角度,将非经典逻辑之一的格值逻辑引入概念格,建立了格值模糊形式背景,通过格结构来刻画对象与属性之间的模糊关系,证明了由蕴涵算子诱导的算子对是伽罗瓦连接,并讨论了相关的一些性质,进而给出了格值模糊概念格的构造算法.格值模糊概念格的建立为模糊性与不可比较性信息的处理提供了可靠的数学工具.     

Regular monoids generated by two galois connections

Consider a Galois connection (α, α) on an ordered setP and a Galois connection (β, β) on the dually ordered set . Arbitrary compositions of these Galois connections form a monoid. In this paper we will examine this monoid. First we prove that it is a regular monoid and then we construct two special Galois connectionsa andb such that every monoid of the above type is a homomorphic image of the monoid generated bya andb, and we give a solution of the word problem of the latter monoid.     

On geometric embedding problems and semiabelian groups

The focus of interest in this paper are geometric Galois realizations of finite groups. A tool to construct such realizations are geometric embedding problems which are introduced. Then the class of semiabelian groups having geometric Galois realizations is studied and characterized in several ways. In particular, a table of small non-semiabelian groups is computed. At the end, geometric Galois realizations of some small non-semiabelian groups are constructed. The author gratefully acknowledges financial support by the Deutsche Forschungsgemeinschaft     

Galois Connections Between Lattices of Preradicals Induced by Adjoint Pairs Between Categories of Modules

We extend the result which establishes that every equivalence between categories of modules induces an isomorphism between the corresponding lattices of preradicals, proving that every adjoint pair between categories of modules induces a Galois connection between the corresponding lattices of preradicals. We prove some general properties of this Galois connection. In particular, given an ideal I of a ring R we study the Galois connection induced by the natural projection R→R/I and we prove that it yields a partition of R-pr into intervals, which are described. We study this partition in the case of the ring \(\mathbb {Z}\) and ideals \(p^{n}\mathbb {Z}\) and use it to locate the idempotent radicals on abelian groups.