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.

Let X be a smooth projective curve of genus > 1 over a field K with function field K(X), let π1(X) be the arithmetic fundamental group of X over K and let GF denote the absolute Galois group of a field F . The section conjecture in Grothendieck’s anabelian geometry says that the sections of the canonical projection π1(X) → GK are (up to conjugation) in one-to-one correspondence with the K-ratio...

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...

Let S be the left R-bialgebroid of a depth two extension with cen-tralizer R. We show that the left endomorphism ring of depth two extension, not necessarily balanced, is a left S-Galois extension of A op. Looking to examples of depth two, we establish that a Hopf subalgebra is normal if and only if it is a Hopf-Galois extension. We find a class of examples of the alternative Hopf algebroids in...

RÉSUMÉ – Organisation de données probabilistes par des treillis de Galois Dans cet article, nous nous intéressons à l’organisation de données probabilistes par des treillis de Galois. Deux correspondances de Galois sont établies, en définissant de façon appropriée les opérateurs de généralisation et d'extension. Ces correspondances permettent de construire deux treillis, appelés treillis de l’u...

Let A and B be R-algebras with automorphism groups G and H respectively. Denote the order of G by n and the order of H by m for some integers n and m. Assume n and m are invertible in R. Then, A⊗R B is a Galois R-algebra with Galois group G × H if and only if A and B are Galois R-algebras with Galois groups G and H respectively. Thus an equivalent condition for a central Galois algebra in terms...

We construct explicitly APF extensions of finite extensions of Qp for which the Galois group is not a p-adic Lie group and which do not have any open subgroup with Zp-quotient. Let K be a finite extension of Qp and L/K be a Galois totally ramified pro-pextension. If the Galois group of L/K is a p-adic Lie group it is known from Sen ([Sen]) that the sequence of upper ramification breaks of L/K i...

We prove a direct image theorem stating that the direct image of a Galois formula by a morphism of difference schemes is equivalent to a Galois formula modulo the theory ACFA of existentially closed difference fields. As a consequence, we obtain an effective quantifier elimination procedure and a precise algebraic-geometric description of definable sets over existentially closed difference fiel...

Coalgebra-Galois extensions generalise Hopf-Galois extensions, which can be viewed as non-commutative torsors. In this paper it is analysed when a coalgebra-Galois extension is a separable, split, or strongly separable extension.

Galois is a domain specific language supported by the Galculator interactive proof-assistant prototype. Galculator uses an equational approach based on Galois connections with indirect equality as an additional inference rule. Galois allows for the specification of different theories in a point-free style by using fork algebras, an extension of relation algebras with expressive power of first-o...