Definition
Consider a partially ordered set (P, ≤) that is a complete lattice. Then P is a complete Heyting algebra if any of the following equivalent conditions hold:
- P is a Heyting algebra, i.e. the operation ( x ∧ − ) 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 P is meet continuous, i.e. the meet operations ( x ∧ − ) are Scott continuous for all x in P.
Read more about this topic: Complete Heyting Algebra
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