A frame is a complete Heyting algebra; in particular it is pseudocomplemented and we have the formula
a^* = \bigvee{x : x \wedge a = 0}.
Dually, in a coframe we have the formula for the supplement
a^# = \bigwedge{x : x v a = 1}. (#)
Somewhat surprisingly, under suitable (and very weak) conditions the formula (#) holds for the pseudocomplement in a frame as well, and more generally (again under suitable conditions) we can borrow the formula for the difference (coHeyting operation)
b \ a = \bigvwedge{x : x v a = 1; x \ge b}
for the Heyting operation a>b. The conditions under which this holds true will be analyzed. As a corollary, one has the theorem on complementarity of the linear elements. The complete lattice S(L) of sublocales of a frame is a coframe. A frame is said to be scattered if S(L) is a frame (in view of the formula above this happens iff it is a Boolean algebra). An aspect of scatteredness concerning general meetrepresentation of sublocales will be discussed.
