Implication algebras

Ivan Chajda

Discussiones Mathematicae - General Algebra and Applications (2006)

  • Volume: 26, Issue: 2, page 141-153
  • ISSN: 1509-9415

Abstract

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We introduce the concepts of pre-implication algebra and implication algebra based on orthosemilattices which generalize the concepts of implication algebra, orthoimplication algebra defined by J.C. Abbott [2] and orthomodular implication algebra introduced by the author with his collaborators. For our algebras we get new axiom systems compatible with that of an implication algebra. This unified approach enables us to compare the mentioned algebras and apply a unified treatment of congruence properties.

How to cite

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Ivan Chajda. "Implication algebras." Discussiones Mathematicae - General Algebra and Applications 26.2 (2006): 141-153. <http://eudml.org/doc/276865>.

@article{IvanChajda2006,
abstract = {We introduce the concepts of pre-implication algebra and implication algebra based on orthosemilattices which generalize the concepts of implication algebra, orthoimplication algebra defined by J.C. Abbott [2] and orthomodular implication algebra introduced by the author with his collaborators. For our algebras we get new axiom systems compatible with that of an implication algebra. This unified approach enables us to compare the mentioned algebras and apply a unified treatment of congruence properties.},
author = {Ivan Chajda},
journal = {Discussiones Mathematicae - General Algebra and Applications},
keywords = {implication algebra; pre-implication algebra; orthoimplication algebra; orthosemilattice; congruence kernel},
language = {eng},
number = {2},
pages = {141-153},
title = {Implication algebras},
url = {http://eudml.org/doc/276865},
volume = {26},
year = {2006},
}

TY - JOUR
AU - Ivan Chajda
TI - Implication algebras
JO - Discussiones Mathematicae - General Algebra and Applications
PY - 2006
VL - 26
IS - 2
SP - 141
EP - 153
AB - We introduce the concepts of pre-implication algebra and implication algebra based on orthosemilattices which generalize the concepts of implication algebra, orthoimplication algebra defined by J.C. Abbott [2] and orthomodular implication algebra introduced by the author with his collaborators. For our algebras we get new axiom systems compatible with that of an implication algebra. This unified approach enables us to compare the mentioned algebras and apply a unified treatment of congruence properties.
LA - eng
KW - implication algebra; pre-implication algebra; orthoimplication algebra; orthosemilattice; congruence kernel
UR - http://eudml.org/doc/276865
ER -

References

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  1. [1] J.C. Abbott, Semi-Boolean algebra, Matem. Vestnik 4 (1967), 177-198. Zbl0153.02704
  2. [2] J.C. Abbott, Orthoimplication Algebras, Studia Logica 35 (1976), 173-177. Zbl0331.02036
  3. [3] L. Beran, Orthomodular Lattices, Algebraic Approach, Mathematic and its Applications, D. Reidel Publ. Comp., 1985. Zbl0558.06008
  4. [4] I. Chajda and R. Halaš, An Implication in Orthologic, submitted to Intern. J. Theor. Phys. 44 (2006), 735-744. Zbl1104.81017
  5. [5] I. Chajda, R. Halaš and H. Länger, Orthomodular implication algebras, Intern. J. Theor. Phys. 40 (2001), 1875-1884. Zbl0992.06008
  6. [6] G.M. Hardegree, Quasi-implication algebras, Part I: Elementary theory, Algebra Universalis 12 (1981), 30-47. Zbl0497.03049
  7. [7] G.M. Hardegree, Quasi-implication algebras, Part II: Sructure theory, Algebra Universalis 12 (1981), 48-65. Zbl0497.03050
  8. [8] J. Hedliková, Relatively orthomodular lattices, Discrete Math., 234 (2001), 17-38. Zbl0983.06008
  9. [9] M.F. Janowitz, A note on generalized orthomodular lattices, J. Natural Sci. Math. 8 (1968), 89-94. Zbl0169.02104
  10. [10] N.D. Megill and M. Pavičić, Quantum implication algebras, Intern. J. Theor. Phys. 48 (2003), 2825-2840. Zbl1039.81007

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