Orthorings

Ivan Chajda; Helmut Länger

Discussiones Mathematicae - General Algebra and Applications (2004)

• Volume: 24, Issue: 1, page 137-147
• ISSN: 1509-9415

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Abstract

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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.

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Ivan Chajda, and Helmut Länger. "Orthorings." Discussiones Mathematicae - General Algebra and Applications 24.1 (2004): 137-147. <http://eudml.org/doc/287738>.

@article{IvanChajda2004,
abstract = {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.},
author = {Ivan Chajda, Helmut Länger},
journal = {Discussiones Mathematicae - General Algebra and Applications},
keywords = {ortholattice; generalized ortholattice; sectionally complemented lattice; orthoring; arithmetical variety; weakly regular variety; congruence kernel; ideal term; basis of ideal terms; subtractive term; orthorings; ortholattices; congruences; ideals},
language = {eng},
number = {1},
pages = {137-147},
title = {Orthorings},
url = {http://eudml.org/doc/287738},
volume = {24},
year = {2004},
}

TY - JOUR
AU - Ivan Chajda
AU - Helmut Länger
TI - Orthorings
JO - Discussiones Mathematicae - General Algebra and Applications
PY - 2004
VL - 24
IS - 1
SP - 137
EP - 147
AB - 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.
LA - eng
KW - ortholattice; generalized ortholattice; sectionally complemented lattice; orthoring; arithmetical variety; weakly regular variety; congruence kernel; ideal term; basis of ideal terms; subtractive term; orthorings; ortholattices; congruences; ideals
UR - http://eudml.org/doc/287738
ER -

References

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1. [1] G. Birkhoff, Lattice Theory, third edition, AMS Colloquium Publ. 25, Providence, RI, 1979. 
2. [2] I. Chajda, Pseudosemirings induced by ortholattices, Czechoslovak Math. J. 46 (1996), 405-411. Zbl0879.06003
3. [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. Zbl1040.06003
4. [4] I. Chajda, G. Eigenthaler and H. Länger, Congruence Classes in Universal Algebra, Heldermann Verlag, Lemgo 2003. Zbl1014.08001
5. [5] I. Chajda and H. Länger, Ring-like operations in pseudocomplemented semilattices, Discuss. Math. Gen. Algebra Appl. 20 (2000), 87-95. Zbl0968.06004
6. [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. Zbl1116.06012
7. [7] I. Chajda, H. Länger and M. Maczyński, Ring-like structures corresponding to generalized orthomodular lattices, Math. Slovaca 54 (2004), 143-150. Zbl1068.06008
8. [8] G. Dorfer, A. Dvurecenskij and H. Länger, Symmetric difference in orthomodular lattices, Math. Slovaca 46 (1996), 435-444. Zbl0890.06006
9. [9] D. Dorninger, H. Länger and M. Maczyński, The logic induced by a system of homomorphisms and its various algebraic characterizations, Demonstratio Math. 30 (1997), 215-232. Zbl0879.06005
10. [10] D. Dorninger, H. Länger and M. Maczyński, On ring-like structures occurring in axiomatic quantum mechanics, Österr. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. II 206 (1997), 279-289. Zbl0945.03095
11. [11] D. Dorninger, H. Länger and M. Maczyński, On ring-like structures induced by Mackey's probability function, Rep. Math. Phys. 43 (1999), 499-515. Zbl1056.81004
12. [12] D. Dorninger, H. Länger and M. Maczyński, Lattice properties of ring-like quantum logics, Intern. J. Theor. Phys. 39 (2000), 1015-1026. Zbl0967.03055
13. [13] D. Dorninger, H. Länger and M. Maczyński, Concepts of measures on ring-like quantum logics, Rep. Math. Phys. 47 (2001), 167-176. Zbl0980.81009
14. [14] D. Dorninger, H. Länger and M. Maczyński, Ring-like structures with unique symmetric difference related to quantum logic, Discuss. Math. General Algebra Appl. 21 (2001), 239-253. Zbl1014.81003
15. [15] G. Grätzer, General Lattice Theory, second edition, Birkhäuser Verlag, Basel 1998. Zbl0909.06002
16. [16] J. Hedlíková, Relatively orthomodular lattices, Discrete Math. 234 (2001), 17-38. Zbl0983.06008
17. [17] M. F. Janowitz, A note on generalized orthomodular lattices, J. Natur. Sci. Math. 8 (1968), 89-94. Zbl0169.02104
18. [18] H. Länger, Generalizations of the correspondence between Boolean algebras and Boolean rings to orthomodular lattices, Tatra Mt. Math. Publ. 15 (1998), 97-105. Zbl0939.03075
19. [19] H. Werner, A Mal'cev condition for admissible relations, Algebra Universalis 3 (1973), 263. Zbl0276.08004

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