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Zbl 0665.06006
Rav, Yehuda
Semiprime ideals in general lattices.
(English)
[J] J. Pure Appl. Algebra 56, No.2, 105-118 (1989). ISSN 0022-4049

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. \par 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). \par 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 $F\subset Q.$ \par 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.
[G.Călugăreanu]
MSC 2000:
*06B10 Ideals, etc. (lattices)

Keywords: coherent local lattice; prime separation; Ultrafilter Principle; semiprime ideal

Cited in: Zbl 1155.06003

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