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Prime \(\alpha\)-ideals in a 0-distributive lattice. (English) Zbl 0595.06010
A lattice L with 0 is said to be 0-distributive if a,b,c\(\in L\), \(a\wedge b=0=a\wedge c\) implies \(a\wedge (b\vee c)=0\). An ideal I of L is called an \(\alpha\)-ideal if \(x\in I\Rightarrow (x)^{**}\subseteq I\). The author introduces a topology on the set of all prime \(\alpha\)-ideals of a 0-distributive lattice L. He proves that the topology is \(T_ 0\) and that the compact open sets form a base for the open sets. He also derives some other properties of the topology and uses them to obtain some equivalent conditions for the space of minimal prime ideals of L with the hull kernel topology to be compact and discrete.

MSC:
06B10 Lattice ideals, congruence relations
06B30 Topological lattices
54D10 Lower separation axioms (\(T_0\)–\(T_3\), etc.)
54D30 Compactness
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