Complete Lattice: Galois connection

Complete Lattice: Galois connection
“a Galois connection is a particular correspondence (typically) between two partially ordered sets (posets).”
Further interesting examples for Galois connections are described in the article on completeness properties. Roughly speaking, it turns out that the usual functions ∨ and ∧ are lower and upper adjoints to the diagonal map X → X × X. The least and greatest elements of a partial order are given by lower and upper adjoints to the unique function X → {1}. Going further, even complete lattices can be characterized by the existence of suitable adjoints. These considerations give some impression of the ubiquity of Galois connections in order theory.

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.
Galois connections also provide an interesting class of mappings between posets which can be used to obtain categories of posets. Especially, it is possible to compose Galois connections: given Galois connections ( f ∗,  f∗) between posets A and B and (g∗, g∗) between B and C, the composite (g∗ ∘  f ∗,  f∗ ∘ g∗) is also a Galois connection. When considering categories of complete lattices, this can be simplified to considering just mappings preserving all suprema (or, alternatively, infima). Mapping complete lattices to their duals, these categories display auto duality, that are quite fundamental for obtaining other duality theorems. More special kinds of morphisms that induce adjoint mappings in the other direction are the morphisms usually considered for frames (or locales).

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