Particular correspondence between two partially ordered sets
In mathematics, especially in order theory, a Galois connection is a particular correspondence (typically) between two partially ordered sets (posets). Galois connections find applications in various mathematical theories. They generalize the fundamental theorem of Galois theory about the correspondence between subgroups and subfields, discovered by the French mathematician Évariste Galois.
A Galois connection can also be defined on preordered sets or classes; this article presents the common case of posets.
The literature contains two closely related notions of "Galois connection". In this article, we will refer to them as (monotone) Galois connections and antitone Galois connections.
A Galois connection is rather weak compared to an order isomorphism between the involved posets, but every Galois connection gives rise to an isomorphism of certain sub-posets, as will be explained below.
The term Galois correspondence is sometimes used to mean a bijective Galois connection; this is simply an order isomorphism (or dual order isomorphism, depending on whether we take monotone or antitone Galois connections).
Galois theory about the correspondence between subgroups and subfields, discovered by the French mathematician Évariste Galois. A Galoisconnection can...
mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental...
equivalently characterized as the lower adjoint part of a unique Galoisconnection. For any pair of preorders P and Q, these are given by pairs of monotone...
Galois theory is a result that describes the structure of certain types of field extensions in relation to groups. It was proved by Évariste Galois in...
}}Y^{c}:=M\backslash Y} . From the above algebras, there exist different types of Galoisconnections, e.g., (1) X ⊆ Y I {\displaystyle X\subseteq Y^{I}} ⟺ Y ⊆ X I {\displaystyle...
Ehresmann connection, gives a manner for differentiating sections of a general fibre bundle Electrical connection, allows the flow of electrons Galois connection...
associative operators. The lower and upper adjoints in a (monotone) Galoisconnection, L and G are quasi-inverses of each other; that is, LGL = L and GLG...
way from a suitable Galoisconnection. The Galoisconnection is not uniquely determined by the closure operator. One Galoisconnection that gives rise to...
to its inverse Adjoint equation The upper and lower adjoints of a Galoisconnection in order theory The adjoint of a differential operator with general...
elaborate type of functions is given by so-called Galoisconnections. Monotone Galoisconnections can be viewed as a generalization of order-isomorphisms...
equivalently, its transitive closure is antisymmetric. Adjoint. See Galoisconnection. Alexandrov topology. For a preordered set P, any upper set O is Alexandrov-open...
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any...
mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups...
rephrase the above definition in terms of the existence of suitable Galoisconnections between related posets — an approach of special interest for category...
graph, and its special case the bipartite double cover Covering group Galoisconnection Quotient space (topology) Hatcher, Allen (2002). Algebraic topology...
orthogonal complement generalizes to the annihilator, and gives a Galoisconnection on subsets of the inner product space, with associated closure operator...
subgroups of a quotient group. More generally, there is a monotone Galoisconnection ( f ∗ , f ∗ ) {\displaystyle (f^{*},f_{*})} between the lattice of...
for Galois groups, the real interest lies often in refining a correspondence to a duality (i.e. antitone order isomorphism). A treatment of Galois theory...
of irreducible closed subsets. This follows immediately from the Galoisconnection between ideals of R and closed subsets of Spec(R) and the observation...
and centrally symmetric spherical polyhedra can be extended to a Galoisconnection including all spherical polyhedra (not necessarily centrally symmetric)...
closure of V is equal to V⊥⊥. The orthogonal complement is thus a Galoisconnection on the partial order of subspaces of a Hilbert space. In general,...
of the ideal generated by S. In more abstract language, there is a Galoisconnection, giving rise to two closure operators; they can be identified, and...
{\displaystyle V\approx V^{**}} . In particular, forming the annihilator is a Galoisconnection on the lattice of subsets of a finite-dimensional vector space. If...
October 2010. Cousot, P.; Cousot, R. (August 1992). "Comparing the GaloisConnection and Widening / Narrowing Approaches to Abstract Interpretation" (PDF)...
let Al denote the set of lower bounds of A. (These operators form a Galoisconnection.) Then the Dedekind–MacNeille completion of S consists of all subsets...
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