Galois connection

The notion of a Galois connection has its root in Galois theoryMathworldPlanetmath. By the fundamental theorem of Galois theory (, there is a one-to-one correspondence between the intermediate fields between a field L and its subfieldMathworldPlanetmath F (with appropriate conditions imposed on the extensionPlanetmathPlanetmathPlanetmath L/F), and the subgroups of the Galois group Gal(L/F) such that the bijection is inclusion-reversing:

Gal(L/F)He iff FLHL, and
FKL iff Gal(L/F)Gal(L/K)e.

If the languagePlanetmathPlanetmath of Galois theory is distilled from the above paragraph, what remains reduces to a more basic and general concept in the theory of ordered-sets:

Definition. Let (P,P) and (Q,Q) be two posets. A Galois connection between (P,P) and (Q,Q) is a pair of functions f:=(f*,f*) with f*:PQ and f*:QP, such that, for all pP and qQ, we have

f*(p)Qq iff pPf*(q).

We denote a Galois connection between P and Q by PfQ, or simply PQ.

If we define P on P by aPb iff bPa, and define Q on Q by cQd iff dQc, then (P,P) and (Q,Q) are posets, (the duals of (P,P) and (Q,Q)). The existence of a Galois connection between (P,P) and (Q,Q) is the same as the existence of a Galois connection between (Q,Q) and (P,P). In short, we say that there is a Galois connection between P and Q if there is a Galois connection between two posets S and T where P and Q are the underlying sets (of S and T respectively). With this, we may say without confusion that “a Galois connection exists between P and Q iff a Galois connection exists between Q and P”.


  1. 1.

    Since f*(p)Qf*(p) for all pP, then by definition, pPf*f*(p). Alternatively, we can write

    1PPf*f*, (1)

    where 1P stands for the identity map on P. Similarly, if 1Q is the identity map on Q, then

    f*f*Q1Q. (2)
  2. 2.

    Suppose aPb. Since bPf*f*(b) by the remark above, aPf*f*(b) and so by definition, f*(a)Qf*(b). This shows that f* is monotone. Likewise, f* is also monotone.

  3. 3.

    Now back to Inequality (1), 1PPf*f* in the first remark. Applying the second remark, we obtain

    f*Qf*f*f*. (3)

    Next, according to Inequality (2), f*f*(q)Qq for any qQ, it is true, in particular, when q=f*(p). Therefore, we also have

    f*f*f*Qf*. (4)

    Putting Inequalities (3) and (4) together we have

    f*f*f*=f*. (5)


    f*f*f*=f*. (6)
  4. 4.

    If (f,g) and (f,h) are Galois connections between (P,P) and (Q,Q), then g=h. To see this, observe that pPg(q) iff f(p)Qq iff pPh(q), for any pP and qQ. In particular, setting p=g(q), we get g(q)Ph(q) since g(q)Pg(q). Similarly, h(q)Pg(q), and therefore g=h. By a similarly argumentPlanetmathPlanetmath, if (g,f) and (h,f) are Galois connections between (P,P) and (Q,Q), then g=h. Because of this uniqueness property, in a Galois connection f=(f*,f*), f* is called the upper adjoint of f* and f* the lower adjoint of f*.


  • The most famous example is already mentioned in the first paragraph above: let L is a finite-dimensional Galois extensionMathworldPlanetmath of a field F, and G:=Gal(L/F) is the Galois group of L over F. If we define

    • a.

      P={KK is a field such that FKL}, with P=,

    • b.

      Q={HH is a subgroup of G}, with Q=,

    • c.

      f*:PQ by f*(K)=Gal(L/K), and

    • d.

      f*:QP by f*(H)=LH, the fixed field of H in L.

    Then, by the fundamental theorem of Galois theory, f* and f* are bijections, and (f*,f*) is a Galois connection between P and Q.

  • Let X be a topological spaceMathworldPlanetmath. Define P be the set of all open subsets of X and Q the set of all closed subsets of X. Turn P and Q into posets with the usual set-theoretic inclusion. Next, define f*:PQ by f*(U)=U¯, the closureMathworldPlanetmathPlanetmath of U, and f*:QP by f*(V)=int(V), the interior of V. Then (f*,f*) is a Galois connection between P and Q. Incidentally, those elements fixed by f*f* are precisely the regular open sets of X, and those fixed by f*f* are the regular closed sets.

Remark. The pair of functions in a Galois connection are order preserving as shown above. One may also define a Galois connection as a pair of maps f*:PQ and f*:QP such that f*(p)Qq iff f*(q)Pp, so that the pair f*,f* are order reversing. In any case, the two definitions are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath in that one may go from one definition to another, (simply exchange Q with Q, the dual ( of Q).


  • 1 T.S. Blyth, Lattices and Ordered Algebraic StructuresPlanetmathPlanetmath, Springer, New York (2005).
  • 2 B. A. Davey, H. A. Priestley, Introduction to Lattices and Order, 2nd Edition, Cambridge (2003)
Title Galois connection
Canonical name GaloisConnection
Date of creation 2013-03-22 15:08:15
Last modified on 2013-03-22 15:08:15
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 13
Author CWoo (3771)
Entry type Definition
Classification msc 06A15
Synonym Galois correspondence
Synonym Galois connexion
Related topic InteriorAxioms
Related topic AdjointFunctor
Defines upper adjoint
Defines lower adjoint