Galois Connection - Existence and Uniqueness of Galois Connections

Existence and Uniqueness of Galois Connections

Another important property of Galois connections is that lower adjoints preserve all suprema that exist within their domain. Dually, upper adjoints preserve all existing infima. From these properties, one can also conclude monotonicity of the adjoints immediately. The adjoint functor theorem for order theory states that the converse implication is also valid in certain cases: especially, any mapping between complete lattices that preserves all suprema is the lower adjoint of a Galois connection.

In this situation, an important feature of Galois connections is that one adjoint uniquely determines the other. Hence one can strengthen the above statement to guarantee that any supremum-preserving map between complete lattices is the lower adjoint of a unique Galois connection. The main property to derive this uniqueness is the following: For every x in A, f ∗(x) is the least element y of B such that x ≤ f _∗(y). Dually, for every y in B, f _∗(y) is the greatest x in A such that f ∗(x) ≤ y. The existence of a certain Galois connection now implies the existence of the respective least or greatest elements, no matter whether the corresponding posets satisfy any completeness properties. Thus, when one adjoint of a Galois connection is given, the other can be defined via this property. On the other hand, some arbitrary function f is a lower adjoint if and only if each set of the form { x in A | f(x) ≤ b }, b in B, contains a greatest element. Again, this can be dualized for the upper adjoint.