On interval homogeneous orthomodular lattices
Anna De Simone, Mirko Navara, Pavel Pták · Czech digital mathematics library · 2001
An orthomodular lattice L is said to be interval homogeneous (resp. centrally interval homogeneous) if it is oe-complete and satisfies the following property: Whenever L is isomorphic to an interval, [a; b], in L then L is isomorphic to each interval [c; d] with c a and d b (resp. the same condition as above only under the assumption that all elements a, b, c, d are central in L). Let us denote by Inthom (resp. Inthomc) the class of all interval homogeneous orthomodular lattices (resp. centrally interval homogeneous orthomodular lattices). We first show that the class Inthom is considerably large --- it contains any Boolean oe-algebra, any block-finite oe-complete orthomodular lattice, any Hilbert space projection lattice and several other examples. Then we prove that L belongs to Inthom exactly when the Cantor-Bernstein-Tarski theorem holds in L. This makes it desirable to know whether there exist oe-complete orthomodular lattices which do not belong to Inthom. Such ex...