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On Boolean ranges of Banaschewski functions. (English) Zbl 1469.06010

Summary: We construct a countable lattice \(\mathcal S\) isomorphic to a bounded sublattice of the subspace lattice of a vector space with two non-iso-morphic maximal Boolean sublattices. We represent one of them as the range of a Banaschewski function and we prove that this is not the case of the other. Hereby we solve a problem of F. Wehrung. We study coordinatizability of the lattice \(\mathcal S\). We prove that although it does not contain a 3-frame, the lattice \(\mathcal S\) is coordinatizable. We show that the two maximal Boolean sublattices correspond to maximal abelian regular subalgebras of the coordinatizating ring.

MSC:

06B05 Structure theory of lattices
06A15 Galois correspondences, closure operators (in relation to ordered sets)
06C20 Complemented modular lattices, continuous geometries
06D75 Other generalizations of distributive lattices
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