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The role of Halaš identity in orthomodular lattices. (English) Zbl 1320.06008

Basic algebras are algebraic structures of signature \((2,1,0)\) corresponding to bounded lattices in which any principal filter \([p,1]\) admits some antitone involution \(x\mapsto x^p\). The main result shows that an ortholattice having this property is orthomodular if and only if any of its subalgebras satisfies \(x\oplus(x\oplus y)=x\oplus y\) where \(x\oplus y=(x'\vee y)^y\) – the binary operation of the associated basic algebra.

MSC:

06C15 Complemented lattices, orthocomplemented lattices and posets
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References:

[1] Beran, L.: Orthomodular Lattices - Algebraic Approach. Academia & D. Reidel Publ. Comp, Praha & Dordrecht, 1984. · Zbl 0558.06008
[2] Chajda, I.: Basic algebras and their applications, an overview. Proc. of the Salzburg Conference AAA81, Contributions to General Algebra 20, Verlag J. Heyn, Klagenfurt, 2011, 1-10. · Zbl 1280.06004
[3] Chajda, I.: Horizontal sums of basic algebras. Discuss. Math., General Algebra Appl. 29 (2009), 21-33. · Zbl 1194.06005 · doi:10.7151/dmgaa.1149
[4] Chajda, I., Halaš, R., Kühr, J.: Distributive lattices with sectionally antitone involutions. Acta Sci. Math. (Szeged) 71 (2005), 19-33. · Zbl 1099.06006
[5] Chajda, I., Halaš, R., Kühr, J.: Many-valued quantum algebras. Algebra Universalis 60 (2009), 63-90. · Zbl 1219.06013 · doi:10.1007/s00012-008-2086-9
[6] Chajda, I., Halaš, R., Kühr, J.: Semilattice Structures. Heldermann Verlag, Lemgo, Germany, 2007. · Zbl 1117.06001
[7] Kalmbach, G.: Orthomodular Lattices. Academic Press, London-New York, 1983. · Zbl 0528.06012
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