Semiprime ideals in general lattices. (English) Zbl 0665.06006

An ideal of a lattice L is called semiprime if for every x,y,z\(\in L\), whenever \(x\wedge y\in I\) and \(x\wedge z\in I\), then \(x\wedge (y\vee z)\in I\). Semiprime filters are dually defined.
Main Theorem. Let L be a lattice and I an ideal of L. The following conditions are equivalent: (1) I is semiprime. (2) I is the kernel of some homomorphism of L onto a distributive lattice with zero. (3) I is the kernel of a homomorphism of L onto a semiprime lattice (if the zero ideal is semiprime).
The following Birkhoff-Stone prime separation theorem generalization is obtained: Corollary. The following statements are equivalent in Zermelo- Fraenkel set theory (without Choice): (a) The Ultrafilter Principle. (b) If a lattice L contains an ideal I and a filter F which are disjoint and such that either I or F is semiprime, then there exists a partition of L by a prime ideal P and a prime filter \(Q=L-P\) such that \(I\subset P\) and \(F\subset Q.\)
Moreover, the author proves several other results such as: Theorem 4.2. Every semiprime ideal of a lattice is representable as an intersection of prime ideals iff the Ultrafilter Principle holds. Theorem 5.2. A lattice is distributive iff, for every ideal I and filter F of L such that \(I\cap F=\emptyset\), there is an ideal J and a filter G of L such that \(I\subset J\), \(F\subset G\), \(J\cap G=\emptyset\), and either J or G is semiprime.
Reviewer: G.Călugăreanu


06B10 Lattice ideals, congruence relations
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