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 xf (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.

Read more about this topic:  Galois Connection

Famous quotes containing the words existence, uniqueness and/or connections:

    If the heart beguiles itself in its choice [of a wife], and imagination will give excellencies which are not the portion of flesh and blood:Mwhen the dream is over, and we awake in the morning, it matters little whether ‘tis Rachael or Leah,—be the object what it will, as it must be on the earthly side ... of perfection,—it will fall short of the work of fancy, whose existence is in the clouds.
    Laurence Sterne (1713–1768)

    Until now when we have started to talk about the uniqueness of America we have almost always ended by comparing ourselves to Europe. Toward her we have felt all the attraction and repulsions of Oedipus.
    Daniel J. Boorstin (b. 1914)

    The conclusion suggested by these arguments might be called the paradox of theorizing. It asserts that if the terms and the general principles of a scientific theory serve their purpose, i. e., if they establish the definite connections among observable phenomena, then they can be dispensed with since any chain of laws and interpretive statements establishing such a connection should then be replaceable by a law which directly links observational antecedents to observational consequents.
    —C.G. (Carl Gustav)