zbMATH — the first resource for mathematics

Ring-like structures corresponding to generalized orthomodular lattices. (English) Zbl 1068.06008
Summary: Ring-like structures, so-called Boolean pseudorings, that are in a natural bijective correspondence with generalized orthomodular lattices are defined. In contrast to generalized orthomodular lattices, Boolean pseudorings are universal algebras and form a variety which turns out to be arithmetical and congruence regular. Distributive Boolean pseudorings, resp. Boolean pseudorings with an associative addition operation are Boolean rings.

06C15 Complemented lattices, orthocomplemented lattices and posets
06E20 Ring-theoretic properties of Boolean algebras
08A30 Subalgebras, congruence relations
Full Text: EuDML
[1] BERAN L.: Orthomodular Lattices. Algebraic Approach, Academia/Reidel, Prague/Dordrecht, 1984. · Zbl 0431.06008
[2] CHAJDA I.: Pseudosemirings induced by ortholattices. Czechoslovak Math. J. 46 (1996), 405-411. · Zbl 0879.06003
[3] CHAJDA I.-EIGENTHALER G.: A note on orthopseudorings and Boolean quasirings. Österreich. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. II 207 (1998), 83-94. · Zbl 1040.06003
[4] CHAJDA I.-LÄNGER H.: Ring-like operations in pseudocomplemented semilattices. Discuss. Math. Gen. Algebra Appl. 20 (2000), 87-95. · Zbl 0968.06004
[5] DILWORTH R. P.: The structure of relatively complemented lattices. Ann. of Math. (2) 51 (1950), 348-359. · Zbl 0036.01802
[6] DORFER G.: Some properties of congruence relations on orthomodular lattices. Discuss. Math. Gen. Algebra Appl. 21 (2001), 57-66. · Zbl 1011.06011
[7] DORNINGER D.-LÄNGER H.-MACZYNSKI M.: The logic induced by a system of homomorphisms and its various algebraic characterizations. Demonstratio Math. 30 (1997), 215-232. · Zbl 0879.06005
[8] DORNINGER D.-LÄNGER H.-MACZYNSKI M.: On ring-like structures occurring in axiomatic quantum mechanics. Österreich. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. II 206 (1997), 279-289. · Zbl 0945.03095
[9] DORNINGER D.-LÄNGER H.-MACZYNSKI M.: On ring-like structures induced by Mackey’s probability function. Rep. Math. Phys. 43 (1999), 499-515. · Zbl 1056.81004
[10] DORNINGER D.-LÄNGER H.-MACZYNSKI M.: Lattice properties of ring-like quantum logics. Internat. J. Theoret. Phys. 39 (2000), 1015-1026. · Zbl 0967.03055
[11] DORNINGER D.-LÄNGER H.-M4CZYNSKI M.: Concepts of measures on ring-like quantum logics. Rep. Math. Phys. 47 (2001), 167-176. · Zbl 0980.81009
[12] DORNINGER D.-LÄNGER H.-MACZYNSKI M.: Ring-like structures with unique symmetric difference related to quantum logic. Discuss. Math. Gen. Algebra Appl. 21 (2001), 239-253. · Zbl 1014.81003
[13] JANOWITZ M. F.: A note on generalized orthomodular lattices. J. Natur. Sci. Math. 8 (1968), 89-94. · Zbl 0169.02104
[14] KALMBACH G.: Orthomodular Lattices. Academic Press, London, 1983. · Zbl 0528.06012
[15] LÄNGER H.: Generalizations of the correspondence between Boolean algebras and Boolean rings to orthomodular lattices. Tatra Mt. Math. Publ. 15 (1998), 97-105. · Zbl 0939.03075
[16] PTÁK P.-PULMANNOVÁ S.: Orthomodular Structures as Quantum Logics. Kluwer Acad. Publ., Dordrecht, 1991. · Zbl 0743.03039
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.