Left adjoint for Booleanization (Q1272544)

A frame is a complete distributive lattice \(L\) in which \(a\wedge\bigvee_{b\in B}b=\bigvee_{b\in B}(a\wedge b)\) for each \(a\in L\) and \(B\subseteq L\). Boolean frames are precisely complete Boolean lattices. If \(L\) is a frame then the set \({\mathcal B}(L)=\{a\in L/a=a^{**}\}\) (where \(a^*\) is the pseudocomplement of a) of all skeletal elements in \(L\) forms a Boolean frame with respect to the induced order, which is called the Booleanization of \(L\). The scope of this paper is to construct many left adjoints for Booleanization in suitable categories.