Prime and maximal ideals of partially ordered sets

Year, pages: 2006, 1 - 22

We exhibit a broad spectrum of classes of ordered sets with the property that in ZF set theory, the Ultrafilter Principle (UP) is equivalent to the validity of a Prime Ideal Theorem (PIT) or of a Maximal Ideal Theorem (MIT), respectively, in the specified class. Weak forms of (semi-)distributivity together with UP yield the desired Prime Ideal Theorems, while weak forms of complementation are responsible for the corresponding Maximal Ideal Theorems. We also study stronger versions, like the extension or separation by prime and maximal ideals, or meet representations by such ideals. Moreover, we investigate slight variations in the definition of prime ideals, which coincide in the case of lattices, but lead to quite different results in the case of posets. Also, rather small changes of the class of posets under consideration may turn a PIT or MIT that was equivalent to UP in one class into a statement equivalent to the full Axiom of Choice (AC) in another class. For example, in the class of arbitrary lower pseudocomplemented posets, PIT is false, while MIT is equivalent to UP, and MIT for upper pseudocomplemented posets is equivalent to AC. Our results extend many known algebraic, lattice-theoretical or topological facts concerning prime and maximal ideals to the setting of partially ordered sets.

Erné, M. 2006. Prime and maximal ideals of partially ordered sets. In Mathematica Slovaca, vol. 56, no.1, pp. 1-22. 0139-9918.

Erné, M. (2006). Prime and maximal ideals of partially ordered sets. Mathematica Slovaca, 56(1), 1-22. 0139-9918.