Editing Complete Heyting algebra (section)

== Definition ==
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 <math>(x\land\cdot)</math> has a [[Adjoint functors|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 (order theory)|distributivity]] law holds:
::<math>x \land  \bigvee_{s \in S} s = \bigvee_{s \in S} (x \land  s).</math>

* ''P'' is a distributive lattice, i.e., for all ''x'', ''y'' and ''z'' in ''P'', we have
::<math>x \land  ( y \lor z ) = ( x \land  y ) \lor ( x \land  z )</math>
: and the meet operations <math>(x\land\cdot)</math> are [[Scott continuous]] (i.e., preserve the suprema of [[directed set]]s) for all ''x'' in ''P''.

The entailed definition of [[Heyting implication]] is <math>a\to b=\bigvee\{c \mid a\land c\le b\}.</math>

Using a bit more category theory, we can equivalently define a frame to be a [[cocomplete]] [[cartesian closed]] [[poset]].