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.


06C15 Complemented lattices, orthocomplemented lattices and posets
Full Text: Link


[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
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.