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Concept
Complete Heyting algebra
Résumé
In mathematics, especially in order theory, a complete Heyting algebra is a Heyting algebra that is complete as a lattice. Complete Heyting algebras are the of three different ; the category CHey, the category Loc of locales, and its , the category Frm of frames. Although these three categories contain the same objects, they differ in their morphisms, and thus get distinct names. Only the morphisms of CHey are homomorphisms of complete Heyting algebras.
Locales and frames form the foundation of pointless topology, which, instead of building on point-set topology, recasts the ideas of general topology in categorical terms, as statements on frames and locales.
Consider a partially ordered set (P, ≤) that is a complete lattice. Then P is a complete Heyting algebra or frame if any of the following equivalent conditions hold:
P is a Heyting algebra, i.e. the operation has a right adjoint (also called the lower adjoint of a (monotone) Galois connection), for each element x of P.
For all elements x of P and all subsets S of P, the following infinite distributivity law holds:
P is a distributive lattice, i.e., for all x, y and z in P, we have
and the meet operations are Scott continuous (i.e., preserve the suprema of directed sets) for all x in P.
The entailed definition of Heyting implication is
Using a bit more category theory, we can equivalently define a frame to be a cocomplete cartesian closed poset.
The system of all open sets of a given topological space ordered by inclusion is a complete Heyting algebra.
The of the category CHey, the category Frm of frames and the category Loc of locales are complete Heyting algebras. These categories differ in what constitutes a morphism:
The morphisms of Frm are (necessarily monotone) functions that preserve finite meets and arbitrary joins.
The definition of Heyting algebras crucially involves the existence of right adjoints to the binary meet operation, which together define an additional implication operation. Thus, the morphisms of CHey are morphisms of frames that in addition preserves implication.
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