a relation between two partially ordered sets in order theory

Specifically, Galois connection is a relation between two partially ordered sets in order theory [20] .

adjoint functors for the category of preordered sets

Galois connections are adjoint functors for the category of preordered sets.

group theory and field theory

Galois is honored as the first mathematician linking group theory and field theory, with the theory that is now called Galois theory.

In mathematics especially in order theory

In mathematics, especially in order theory, a Galois connection is a particular correspondence (typically) between two partially ordered sets (posets).

a pair of antitone

In this alternative definition, a Galois connection is a pair of antitone, i.e. order-reversing, functions and between two posets and , such that if and only if .

adjoint functors](https://en.wikipedia.org/wiki/Adjoint_functors

Galois connections are [adjoint functors](https://en.wikipedia.org/wiki/Adjoint_functors) for the category of preordered sets.

the upper adjoint of the function that embeds the integers into the reals

In the language of order theory, the floor function is a residuated mapping, that is, part of a Galois connection: a Galois connection is the upper adjoint of the function that embeds the integers into the reals.

theory of polynomial equations and group theory

Galois theory gives a beautiful connection between the theory of polynomial equations and group theory.

an adjunction

Then $${\sf Par\dashv Chi}$$ since we have a (covariant) Galois connection between the poset $\big(\mathcal{P}(\mathscr{X}),\subseteq\big)$ and itself given by the above functions (your parent's children includes you and your children's parents also includes you*), and a covariant Galois connection is an adjunction.

correspondence between subgroups and subfields investigated in Galois theory

Galois connections generalize the correspondence between subgroups and subfields investigated in Galois theory (named after the French mathematician Évariste Galois).

a more abstract duality relationship , known as a Galois connection

In the absence of quasilinearity, a more abstract duality relationship, known as a Galois connection, takes the role of (generalized) conjugate duality.

to closure operators and to inverse order-preserving bijections between the corresponding closed elements

Any Galois connection gives rise to closure operators and to inverse order-preserving bijections between the corresponding closed elements.

mathematical data analysis

That field uses Galois connections for mathematical data analysis.

abstract framework

Galois connections form the abstract framework not only for classical and modern Galois theory, involving groups, fields and rings, but also for many other algebraic, topological, ordertheoretical, categorical and logical theories.

a more abstract duality relationship, known as a Galois connection

In the absence of quasilinearity, a more abstract duality relationship, known as a Galois connection, takes the role of (generalized) conjugate duality.

a contravariant adjunction , between posets P,QP

Well, to me the phrase “Galois connection” mainly connotes a contravariant adjunction between posets P,QP, Q.

a slicer (that goes from the part of a trace you care about to the parts of a program you should care about)

is essential: the Galois connection specifies a slicer (that goes from the part of a trace you care about to the parts of a program you should care about) by relating a program to an evaluator (that goes from the part of the program you know about to the parts of the trace you can know about).

a nice way of thinking about the fundamental theorem of Galois theory, and the Nullstellsatz

Galois connections give a nice way of thinking about the fundamental theorem of Galois theory, and the Nullstellsatz.

a many-valued Galois connection with respect to similarity between attribute values in a many-valued context

We define a many-valued Galois connection with respect to similarity between attribute values in a many-valued context.

an order reversing correspondence between the posets which is a lattice dual isomorphism between the posets of 'closed' elements

The Galois Connection is then an order reversing correspondence between the posets which is a lattice dual isomorphism between the posets of 'closed' elements.