Galois Connection - Closure Operators and Galois Connections

Closure Operators and Galois Connections

The above findings can be summarized as follows: for a Galois connection, the composite f _∗f ∗ is monotone (being the composite of monotone functions), inflationary, and idempotent. This states that f _∗f ∗ is in fact a closure operator on A. Dually, f ∗f _∗ is monotone, deflationary, and idempotent. Such mappings are sometimes called kernel operators. In the context of frames and locales, the composite f _∗f ∗ is called the nucleus induced by f. Nuclei induce frame homomorphisms; a subset of a locale is called a sublocale if it is given by a nucleus.

Conversely, any closure operator c on some poset A gives rise to the Galois connection with lower adjoint f ∗ being just the corestriction of c to the image of c (i.e. as a surjective mapping the closure system c(A)). The upper adjoint f _∗ is then given by the inclusion of c(A) into A, that maps each closed element to itself, considered as an element of A. In this way, closure operators and Galois connections are seen to be closely related, each specifying an instance of the other. Similar conclusions hold true for kernel operators.

The above considerations also show that closed elements of A (elements x with f _∗(f ∗(x)) = x) are mapped to elements within the range of the kernel operator f ∗ f _∗, and vice versa.