We introduce Galois corings, and give a survey of properties that have been obtained so far. The Definition is motivated using descent theory, and we show that classical Galois theory, Hopf-Galois theory and coalgebra Galois theory can be obtained as a special case.

Generalizing Euclidean inner product and Hermitian inner product, we introduce Galois inner products, and study the Galois self-dual constacyclic codes in a very general setting by a uniform method. The conditions for existence of Galois self-dual and isometrically Galois self-dual constacyclic codes are obtained. As consequences, the results on self-dual, iso-dual and Hermitian self-dual const...

We introduce a new device in the study of abstract elementary classes (AECs): Galois Morleyization, which consists in expanding the models of the class with a relation for every Galois (orbital) type of length less than a fixed cardinal κ. We show: Theorem 0.1 (The semantic-syntactic correspondence). An AEC K is fully (< κ)-tame and type short if and only if Galois types are syntactic in the Ga...

Substitution Box or S-Box had been generated using 4-bit Boolean Functions (BFs) for Encryption and Decryption Algorithm of Lucifer and Data Encryption Standard (DES) in late sixties and late seventies respectively. The SBox of Advance Encryption Standard have also been generated using Irreducible Polynomials over Galois field GF(2 8 ) adding an additive constant in early twenty first century. ...

The theory of concept (or Galois) lattices provides a natural and formal setting in which to discover and represent concept hierarchies. In this paper we present a system, GALOIS, which is able to determine the concept lattice corresponding to a given set of objects. GALOIS is incremental and relatively efficient, the time complexity of each update ranging from O(n) to O(n2) where n is the numb...

We present a general formula for the intent–extent mappings of a Galois lattice generated by individual descriptions which lie in any arbitrary lattice. The formulation is unique if a natural maximality condition is required. This formulation yields, as particular cases, formal concept binary Galois lattices of Wille, those de0ned by Brito or Blyth–Janowitz, as well as fuzzy or stochastic Galoi...

This work presents a difference geometric approach to the strongly normal Galois theory of difference equations. In this approach, a system of ordinary difference equations is encoded in a difference extension, and the Galois groups are group schemes of finite type over the constants. The Galois groups need neither be linear nor reduced. The main result is a characterization of the extensions t...

We study the basic Galois connection induced by the ”satisfaction” relation between external functions An → B defined on a set A and valued in a possibly different set B, and ordered pairs (R, S) of relations R ⊆ Am and S ⊆ Bm, called relational constraints. We represent the induced Galois operators as compositions of closure operators associated with necessary and sufficient conditions describ...

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.

Picard-Vessiot rings are present in many settings like differential Galois theory, difference Galois theory and Galois theory of Artinian simple module algebras. In this article we set up an abstract framework in which we can prove theorems on existence and uniqueness of Picard-Vessiot rings, as well as on Galois groups corresponding to the Picard-Vessiot rings. As the present approach restrict...