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On the permanence properties of interval homogeneous orthomodular lattices. (English) Zbl 1077.06005
An orthomodular lattice \(L\) is called interval homogeneous if (i) it is \(\sigma \)-complete, and (ii) whenever it is isomorphic to some of its intervals \([a,b]\), then it is isomorphic to each of its intervals \([c,d]\supseteq [a,b]\). The class of all such lattices is denoted by Inthom. The authors remark that their investigation of this class was inspired by some results of Cantor-Bernstein type. A \(\sigma \)-complete orthomodular lattice is denoted as \(\sigma \)-OML.
Sample results: Inthom is a proper class (in fact, a stronger result concerning a subclass of Inthom is proved). The authors find sufficient conditions under which a \(\sigma \)-OML belongs to Inthom. Each \(\sigma \)-OML is a \(\sigma \)-epimorphic image of an element of Inthom. Further, each \(\sigma \)-OML is a \(\sigma \)-epimorphic image of a \(\sigma \)-OML which does not belong to Inthom. Analogous results are obtained when instead of \(\sigma \)-homomorphic images we deal with sub-\(\sigma \)-algebras.

MSC:
06C15 Complemented lattices, orthocomplemented lattices and posets
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