Some new intuitionistic equivalents of Zorn's Lemma

In classical set theory, Zorn's Lemma is equivalent to the axiom of choice and a host of other principles and theorems. But in intuitionistic set theory (IZF), in which the law of excluded middle is not assumed, the situation is quite different. (A presentation of IZF may be found in Chapter VIII of [3].) Here, Zorn's lemma turns out to be remarkably weak: not only does it fail to imply the axiom of choice, but one cannot even prove from it, for example, the Boolean prime ideal theorem or the Stone representation theorem. (This is because, as shown in [4], Zorn's lemma has no nonconstructive purely logical consequences, while both the axiom of choice and the Stone representation theorem imply the law of excluded middle, and the Boolean prime ideal theorem implies the nonconstructive form of de Morgan's law: see [5].) In fact, the vast majority of the assertions intuitionistically provable from Zorn's lemma make explicit mention of the notion of maximality: for example, the existence of maximal chains in partially ordered sets and the maximal ideal theorem for rings. (A conspicuous exception to this is the Sikorski extension theorem for complete Boolean algebras which is intuitionistically derivable from Zorn's lemma: see [4].) In this note two apparently new results are proved, neither of which make explicit reference to maximality—the one a fixpoint theorem for complete lattices, the other a result concerning binary relations—and each is shown to be intuitionistically equivalent to Zorn's lemma. We begin with some