×

On varieties of modular ortholattices that are generated by their finite-dimensional members. (English) Zbl 1306.06006

A modular ortholattice is an algebra \((L,\vee,\wedge,',0,1)\) of type \((2,2,1,0,0)\) such that \((L,\vee,\wedge,0,1)\) is a bounded modular lattice and the identities \(x\vee x'=1\), \(x\wedge x'=0\), \((x')'=x\), \((x\vee y)'=x'\wedge y'\) and \((x\wedge y)'=x'\vee y'\) hold. Let \(\mathcal L\) be a modular ortholattice and \(\mathcal V\) a variety of modular ortholattices. The dimension of \(\mathcal L\) is the cardinality of a maximal chain in \(\mathcal L\) minus one. The fact that \(\mathcal L\) belongs to the variety generated by the finite-dimensional members of \(\mathcal V\) is characterized in two different ways.

MSC:

06C15 Complemented lattices, orthocomplemented lattices and posets
06C20 Complemented modular lattices, continuous geometries
08B15 Lattices of varieties
51A50 Polar geometry, symplectic spaces, orthogonal spaces
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Baer R.: Polarities in finite projective planes, Bull. Amer. Math. Soc. 51, 77-93 (1946) · Zbl 0060.32308 · doi:10.1090/S0002-9904-1946-08506-7
[2] Birkhoff G, von Neumann J.: The logic of quantum mechanics, Ann. of Math. 37, 823-843 (1936) · Zbl 0015.14603 · doi:10.2307/1968621
[3] Bruns G.: Varieties of modular ortholattices, Houston J. Math. 9, 1-7 (1983) · Zbl 0518.06007
[4] Faure C.A., Frölicher A.: Modern Projective Geometry. Kluwer, Dordrecht (2000) · doi:10.1007/978-94-015-9590-2
[5] Grätzer G.: General Lattice Theory, 2nd edn. Birkhäuser, Basel (1998) · Zbl 0385.06015
[6] Harding J.: Decidability of the equational theory of the continuous geometry CG \[({\mathbb{F}}F)\], J. Philos. Logic 42, 461-465 (2013) · Zbl 1345.03022 · doi:10.1007/s10992-013-9270-x
[7] Herrmann C.: On representations of complemented modular lattices with involution, Algebra Universalis 61, 339-364 (2009) · Zbl 1192.06008 · doi:10.1007/s00012-009-0003-5
[8] Herrmann C.: On the equational theory of projection lattices of finite von Neumann factors, J. Symb. Logic 75, 1102-1110 (2010) · Zbl 1205.06005 · doi:10.2178/jsl/1278682219
[9] Herrmann C., Roddy M.S.: Proatomic modular ortholattices: Representation and equational theory, Note di matematica e fisica 10, 55-88 (1999)
[10] Herrmann C., Roddy M.S.: A note on the equational theory of modular ortholattices, Algebra Universalis 44, 165-168 (2000) · Zbl 1014.06006 · doi:10.1007/s000120050178
[11] Herrmann C., Roddy M.S.: Three ultrafilters in a modular logic, Internat. J. Theoret. Phys. 50, 3821-3827 (2011) · Zbl 1250.06002 · doi:10.1007/s10773-011-0902-z
[12] Herrmann C., Roddy M.S.: On geometric representations of modular ortholattices, Algebra Universalis 71, 285-297 (2014) · Zbl 1295.06002 · doi:10.1007/s00012-014-0278-z
[13] Holland Jr., S.S.: The current interest in orthomodular lattices. In: Abbott, J.C. (ed.) Trends in Lattice Theory, pp. 41-126. van Norstrand, New York (1970) · Zbl 1192.06008
[14] Kalmbach, G: MR0699045 (84j:06010) · Zbl 0815.28004
[15] von Neumann J.: Continuous geometries and examples of continuous geometries. Proc. Nat. Acad. Acad. Sci. U.S.A 22, 707-713 (1936) · JFM 62.1103.03 · doi:10.1073/pnas.22.12.707
[16] Roddy M. S.: Varieties of modular ortholattices, Order 3, 405-426 (1987) · Zbl 0618.06004 · doi:10.1007/BF00340782
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.