ISSN 1509-9415 (print version)

ISSN 2084-0373 (electronic version)

Discussiones Mathematicae - General Algebra and Applications

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H-Index: 5

Discussiones Mathematicae - General Algebra and Applications


Discussiones Mathematicae General Algebra and Applications 24(1) (2004) 137-147


Ivan Chajda

Palacký University, Olomouc
Department of Algebra and Geometry
Tomkova 40, 77900 Olomouc, Czech Republic


Helmut Länger

Vienna University of Technology
Institute of Discrete Mathematics and Geometry
Research Unit Algebra
Wiedner Hauptstraß e 8-10, 1040 Vienna, Austria



Certain ring-like structures, so-called orthorings, are introduced which are in a natural one-to-one correspondence with lattices with 0 every principal ideal of which is an ortholattice. This correspondence generalizes the well-known bijection between Boolean rings and Boolean algebras. It turns out that orthorings have nice congruence and ideal properties.

Keywords: ortholattice, generalized ortholattice, sectionally complemented lattice, orthoring, arithmetical variety, weakly regular variety, congruence kernel, ideal term, basis of ideal terms, subtractive term.

2000 Mathematics Subject Classification: 16Y99, 06C15, 81P10.


[1] G. Birkhoff, Lattice Theory, third edition, AMS Colloquium Publ. 25, Providence, RI, 1979.
[2] I. Chajda, Pseudosemirings induced by ortholattices, Czechoslovak Math. J. 46 (1996), 405-411.
[3] I. Chajda and G. Eigenthaler, A note on orthopseudorings and Boolean quasirings, Österr. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. II 207 (1998), 83-94.
[4] I. Chajda, G. Eigenthaler and H. Länger, Congruence Classes in Universal Algebra, Heldermann Verlag, Lemgo 2003.
[5] I. Chajda and H. Länger, Ring-like operations in pseudocomplemented semilattices, Discuss. Math. Gen. Algebra Appl. 20 (2000), 87-95.
[6] I. Chajda and H. Länger, Ring-like structures corresponding to MV-algebras via symmetric difference, Österr. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. II, to appear.
[7] I. Chajda, H. Länger and M. Maczy\'nski, Ring-like structures corresponding to generalized orthomodular lattices, Math. Slovaca 54 (2004), 143-150.
[8] G. Dorfer, A. Dvurecenskij and H. Länger, Symmetric difference in orthomodular lattices, Math. Slovaca 46 (1996), 435-444.
[9] D. Dorninger, H. Länger and M. Maczy\'nski, The logic induced by a system of homomorphisms and its various algebraic characterizations, Demonstratio Math. 30 (1997), 215-232.
[10] D. Dorninger, H. Länger and M. Maczy\'nski, On ring-like structures occurring in axiomatic quantum mechanics, Österr. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. II 206 (1997), 279-289.
[11] D. Dorninger, H. Länger and M. Maczy\'nski, On ring-like structures induced by Mackey's probability function, Rep. Math. Phys. 43 (1999), 499-515.
[12] D. Dorninger, H. Länger and M. Maczy\'nski, Lattice properties of ring-like quantum logics, Intern. J. Theor. Phys. 39 (2000), 1015-1026.
[13] D. Dorninger, H. Länger and M. Maczy\'nski, Concepts of measures on ring-like quantum logics, Rep. Math. Phys. 47 (2001), 167-176.
[14] D. Dorninger, H. Länger and M. Maczy\'nski, Ring-like structures with unique symmetric difference related to quantum logic, Discuss. Math. General Algebra Appl. 21 (2001), 239-253.
[15] G. Grätzer, General Lattice Theory, second edition, Birkhäuser Verlag, Basel 1998.
[16] J. Hedlíková, Relatively orthomodular lattices, Discrete Math. 234 (2001), 17-38.
[17] M. F. Janowitz, A note on generalized orthomodular lattices, J. Natur. Sci. Math. 8 (1968), 89-94.
[18] H. Länger, Generalizations of the correspondence between Boolean algebras and Boolean rings to orthomodular lattices, Tatra Mt. Math. Publ. 15 (1998), 97-105.
[19] H. Werner, A Mal'cev condition for admissible relations, Algebra Universalis 3 (1973), 263.

Received 2 March 2004
Revised 9 June 2004