ABSTRACT
One would expect t h a t the notion of quantaloidal nucleus is m ore general t h a n t h a t of congruence and, in fact, F(G(j)) m a y fail to equal j in d r a m a t i c fashion. For example , let Q be a q u a n ta le and consider the quan t ic nucleus j on P(Q) defined j(A) = (sup A)1 = {b € Q | b < sup A). Then, (x,y) € iff j(x) = j(y) iff x = y, i.e. ©j is the diagonal relation on Q and hence F(Oj) is the ident i ty nucleus on P(Q).
