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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.

06C15 Complemented lattices, orthocomplemented lattices and posets
Full Text: EuDML
[1] BERAN L.: Orthomodular Lattices. Algebraic Approach, Academia/D. Reidel, Praha/Dordrecht, 1984. · Zbl 0558.06008
[2] DE SIMONE A.-MUNDICI D.-NAVARA M.: A : Cantor-Bernstein theorem for a-complete MV-algebras. Czechoslovak Math. J. 53 (128) (2003), 437-447. · Zbl 1024.06003
[3] DE SIMONE A.-NAVARA M.-PTÁK P.: On interval homogeneous orthomodular lattices. Comment. Math. Univ. Carolin. 42 (2001), 23-30. · Zbl 1052.06007
[4] FREYTES H.: An algebraic version of the Cantor-Bernstein-Schroder Theorem. Czechoslovak Math. J. · Zbl 1080.06008
[5] JAKUBÍK J.: A theorem of Cant or-Bernstein type for orthogonally a-complete pseudo MV-algebras. Tatra Mt. Math. Publ. 22 (2002), 91-103.
[6] JENČA G.: A Cant or-Bernstein type theorem for effect algebras. Algebra Universalis 48 (2002), 399-411. · Zbl 1061.06020
[7] KALMBACH G.: Orthomodular Lattices. Academic Press, London, 1983. · Zbl 0528.06012
[8] KALLUS M.-TRNKOVÁ V.: Symmetries and retracts of quantum logics. Internat. J. Theor. Phys. 26 (1987), 1-9. · Zbl 0626.06013
[9] Handbook of Boolean Algebras I. (J. D. Monk, R. Bonnet, North Holland Elsevier Science Publisher B.V., Amsterdam, 1989.
[10] PTÁK P.-PULMANNOVÁ S.: Orthomodular Structures as Quantum Logics. Kluwer, Dordrecht-Boston-London, 1991. · Zbl 0743.03039
[11] TRNKOVÁ V.: Automorphisms and symmetries of quantum logics. Internat. J. Theor. Physics 28 (1989), 1195-1214. · Zbl 0697.03034
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