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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