Terms of use:

The Prime Ideal Theorem is shown to be equivalent with the following two statements (1) Any compact nontrivial quantale has a prime element, and (2) Arty compact normal complete nontrivial distributive lattice has a maximal element. Some another equivalents of the Prime Ideal Theorem are given. The lattice-theoretical investigations of complete lattices equipped with an ad ditional binary operation • which distributes over arbitrary joins in each variable can be traced back to Ward and Dilworth [14]. Such a gadget is called quantale (following C. J. Mulvey). Some topological properties of quantales were obtained by Borceux [4]. The original motivation for this paper was the question whether the Prime Ideal Theorem (every nontrivial distributive lattice has a prime ideal) is strong enough to ensure the existence of a prime element in arbitrary compact quantale, rather than just in compact frames (see [3], [7]). We present some new relationships between this principle and quantale-theoretic conditions, e.g. the existence of a maximal element in arbitrary compact normal frame