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.


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
Full Text: DOI


[1] Banaschewski, B, Totalgeordnete moduln, Arch. Math., 7, 430-440, (1957) · Zbl 0208.03702
[2] Burris, S., Sankappanavar, H.B.: A Course in Universal Algebra. Graduate Texts in Mathematics, vol. 78. Springer, New-York (1981) · Zbl 0478.08001
[3] Engelking, R.: General Topology. Heldermann, Berlin (1989) · Zbl 0684.54001
[4] Goodearl, K.R.: Von Neumann Regular Rings, 2nd edn. Krieger Pub. Co., Malabar, FL (1991) · Zbl 0749.16001
[5] Grätzer, G.: General Lattice Theory, 2nd edn. Birkhäuser, Basel (1998) · Zbl 0909.06002
[6] Grätzer, G; Schmidt, ET, A lattice construction and congruence-preserving extensions, Acta Math. Hung., 66, 275-288, (1995) · Zbl 0842.06008
[7] Grätzer, G; Schmidt, ET, On the independence theorem of related structures for modular (Arguesian) lattices, Stud.Sci. Math. Hungar., 40, 1-12, (2003) · Zbl 1048.06004
[8] Grätzer, G; Wehrung, F, Flat semilattices, Colloq. Math., 79, 185-191, (1999) · Zbl 0922.06006
[9] Grätzer, G; Wehrung, F, The \(M_3 [D]\) construction and \(n\)-modularity, Algebra Univers., 41, 87-114, (1999) · Zbl 0965.06008
[10] Grätzer, G; Wehrung, F, A new lattice construction: the box product, J. Algebra, 221, 5893-5919, (1999) · Zbl 0961.06005
[11] Grätzer, G; Wehrung, F, Proper congruence-preserving extension of lattices, Acta Math. Hungar., 85, 169-179, (1999) · Zbl 0988.06002
[12] Grätzer, G; Wehrung, F, Tensor product and transferability of semilattices, Can. J. Math., 51, 792-815, (1999) · Zbl 0941.06010
[13] Grätzer, G; Wehrung, F, Tensor product and semilattices with zero, revisited, J. Pure Appl. Algebra, 147, 273-301, (2000) · Zbl 0945.06003
[14] Grätzer, G; Wehrung, F, A survey of tensor product and related structures in two lectures, Algebra Univers., 45, 117-143, (2001) · Zbl 0981.06002
[15] Jónsson, B, Representations of complemented modular lattices, Trans. Am. Math. Soc., 60, 64-94, (1960) · Zbl 0101.02204
[16] Murphy, G.J.: \(C^⁎ \)-Algebras and Operator Theory. Academic Press. Inc, London (1990) · Zbl 0714.46041
[17] Ore, Ø, Galois connexions, Trans. Am. Math. Soc., 55, 493-513, (1944) · Zbl 0060.06204
[18] Schmidt, ET, Zur charakterisierung der kongruenzverbände der verbände, Mat. Časopis Sloven. Akad. Vied., 18, 3-20, (1968) · Zbl 0155.35102
[19] Schmidt, ET, Every finite distributive lattice is the congruence lattice of a modular lattice, Algebra Univers., 4, 49-57, (1974) · Zbl 0298.06013
[20] Wehrung, F, Coordinatization of lattices by regular rings without unit and banaschewski functions, Algebra Univers., 64, 49-67, (2010) · Zbl 1205.06006
[21] Wehrung, F, A non-coordinatizable sectionally complemented modular lattice with a large Jónsson four-frame, Adv. Appl. Math., 47, 173-193, (2011) · Zbl 1244.06003
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