Directoids with sectionally antitone involutions and skew MV-algebras

Ivan Chajda; Miroslav Kolařík

Mathematica Bohemica (2007)

  • Volume: 132, Issue: 4, page 407-422
  • ISSN: 0862-7959

Abstract

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It is well-known that every MV-algebra is a distributive lattice with respect to the induced order. Replacing this lattice by the so-called directoid (introduced by J. Ježek and R. Quackenbush) we obtain a weaker structure, the so-called skew MV-algebra. The paper is devoted to the axiomatization of skew MV-algebras, their properties and a description of the induced implication algebras.

How to cite

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Chajda, Ivan, and Kolařík, Miroslav. "Directoids with sectionally antitone involutions and skew MV-algebras." Mathematica Bohemica 132.4 (2007): 407-422. <http://eudml.org/doc/250241>.

@article{Chajda2007,
abstract = {It is well-known that every MV-algebra is a distributive lattice with respect to the induced order. Replacing this lattice by the so-called directoid (introduced by J. Ježek and R. Quackenbush) we obtain a weaker structure, the so-called skew MV-algebra. The paper is devoted to the axiomatization of skew MV-algebras, their properties and a description of the induced implication algebras.},
author = {Chajda, Ivan, Kolařík, Miroslav},
journal = {Mathematica Bohemica},
keywords = {directoid; antitone involution; sectionally switching mapping; MV-algebra; NMV-algebra; WMV-algebra; skew MV-algebra; implication algebra; directoid; antitone involution; sectionally switching mapping},
language = {eng},
number = {4},
pages = {407-422},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Directoids with sectionally antitone involutions and skew MV-algebras},
url = {http://eudml.org/doc/250241},
volume = {132},
year = {2007},
}

TY - JOUR
AU - Chajda, Ivan
AU - Kolařík, Miroslav
TI - Directoids with sectionally antitone involutions and skew MV-algebras
JO - Mathematica Bohemica
PY - 2007
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 132
IS - 4
SP - 407
EP - 422
AB - It is well-known that every MV-algebra is a distributive lattice with respect to the induced order. Replacing this lattice by the so-called directoid (introduced by J. Ježek and R. Quackenbush) we obtain a weaker structure, the so-called skew MV-algebra. The paper is devoted to the axiomatization of skew MV-algebras, their properties and a description of the induced implication algebras.
LA - eng
KW - directoid; antitone involution; sectionally switching mapping; MV-algebra; NMV-algebra; WMV-algebra; skew MV-algebra; implication algebra; directoid; antitone involution; sectionally switching mapping
UR - http://eudml.org/doc/250241
ER -

References

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  2. Congruence Classes in Universal Algebra, Heldermann, Lemgo (Germany), 2003. (2003) MR1985832
  3. Distributive lattices with sectionally antitone involutions, Acta Sci. Math. (Szeged) 71 (2005), 19–33. (2005) MR2160352
  4. 10.1007/s00012-004-1862-4, Algebra Universalis 52 (2004), 377–382. (2004) MR2120523DOI10.1007/s00012-004-1862-4
  5. A non-associative generalization of MV-algebras, (to appear). (to appear) MR2357826
  6. 10.1090/S0002-9947-1958-0094302-9, Trans. Amer. Math. Soc. 88 (1958), 467–490. (1958) MR0094302DOI10.1090/S0002-9947-1958-0094302-9
  7. Algebraic Foundations of Many- Valued Reasoning, Kluwer Acad. Publ., Dordrecht, 2000. (2000) MR1786097
  8. Pseudo MV-algebras, Multiple Valued Log. 6 (2001), 95–135. (2001) MR1817439
  9. Implication reducts of weak MV-algebras, Contributions to General Algebra 18, Verlag Heyn, 2007, to appear. MR2399238
  10. Weak MV-algebras, (to appear). (to appear) MR2399238
  11. 10.1007/BF01190253, Algebra Universalis 27 (1990), 49–69. (1990) MR1025835DOI10.1007/BF01190253
  12. 10.1023/A:1021766309509, Czech. Math. J. 52 (2002), 255–273. (2002) Zbl1012.06012MR1905434DOI10.1023/A:1021766309509

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