Directoids with an antitone involution

Ivan Chajda; Miroslav Kolařík

Commentationes Mathematicae Universitatis Carolinae (2007)

  • Volume: 48, Issue: 4, page 555-569
  • ISSN: 0010-2628

Abstract

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We investigate ⊓ -directoids which are bounded and equipped by a unary operation which is an antitone involution. Hence, a new operation ⊔ can be introduced via De Morgan laws. Basic properties of these algebras are established. On every such an algebra a ring-like structure can be derived whose axioms are similar to that of a generalized boolean quasiring. We introduce a concept of symmetrical difference and prove its basic properties. Finally, we study conditions of direct decomposability of directoids with an antitone involution.

How to cite

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Chajda, Ivan, and Kolařík, Miroslav. "Directoids with an antitone involution." Commentationes Mathematicae Universitatis Carolinae 48.4 (2007): 555-569. <http://eudml.org/doc/250201>.

@article{Chajda2007,
abstract = {We investigate $\sqcap $-directoids which are bounded and equipped by a unary operation which is an antitone involution. Hence, a new operation $\sqcup $ can be introduced via De Morgan laws. Basic properties of these algebras are established. On every such an algebra a ring-like structure can be derived whose axioms are similar to that of a generalized boolean quasiring. We introduce a concept of symmetrical difference and prove its basic properties. Finally, we study conditions of direct decomposability of directoids with an antitone involution.},
author = {Chajda, Ivan, Kolařík, Miroslav},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {directoid; antitone involution; D-quasiring; symmetrical difference; direct decomposition; directoid; antitone involution; D-quasiring; symmetrical difference; direct decomposition},
language = {eng},
number = {4},
pages = {555-569},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Directoids with an antitone involution},
url = {http://eudml.org/doc/250201},
volume = {48},
year = {2007},
}

TY - JOUR
AU - Chajda, Ivan
AU - Kolařík, Miroslav
TI - Directoids with an antitone involution
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2007
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 48
IS - 4
SP - 555
EP - 569
AB - We investigate $\sqcap $-directoids which are bounded and equipped by a unary operation which is an antitone involution. Hence, a new operation $\sqcup $ can be introduced via De Morgan laws. Basic properties of these algebras are established. On every such an algebra a ring-like structure can be derived whose axioms are similar to that of a generalized boolean quasiring. We introduce a concept of symmetrical difference and prove its basic properties. Finally, we study conditions of direct decomposability of directoids with an antitone involution.
LA - eng
KW - directoid; antitone involution; D-quasiring; symmetrical difference; direct decomposition; directoid; antitone involution; D-quasiring; symmetrical difference; direct decomposition
UR - http://eudml.org/doc/250201
ER -

References

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  1. Chajda I., Pseudosemirings induced by ortholattices, Czechoslovak Math. J., 46 (1996), 405-411. (1996) Zbl0879.06003MR1408295
  2. Chajda I., Eigenthaler G., A note on orthopseudorings and Boolean quasirings, Österr. Akad. Wiss. Math.-Natur., Kl., Sitzungsber. II, 207 (1998), 83-94. (1998) Zbl1040.06003MR1749914
  3. Dorfer D., Dvurečenskij A., Länger H., Symmetrical difference in orthomodular lattices, Math. Slovaca 46 (1996), 435-444. (1996) MR1451034
  4. Dorminger D., Länger H., Mączyński M., The logic induced by a system of homomorphisms and its various algebraic characterizations, Demonstratio Math. 30 (1997), 215-232. (1997) MR1446613
  5. Gardner B.J., Parmenter M.M., Directoids and directed groups, Algebra Universalis 33 (1995), 254-273. (1995) Zbl0832.06005MR1318990
  6. Ježek J., Quackenbush R., Directoids: algebraic models of up-directed sets, Algebra Universalis 27 (1990), 49-69. (1990) MR1025835
  7. Kopytov V.M., Dimitrov Z.I., On directed groups, Siberian Math. J. 30 (1989), 895-902; (Russian original: Sibirsk. Mat. Zh. 30 (1988), no. 6, 78-86). (1989) Zbl0714.06007MR1043436
  8. Leutola K., Nieminen J., Posets and generalized lattices, Algebra Universalis 16 (1983), 344-354. (1983) Zbl0514.06003MR0695054
  9. Nieminen J., On distributive and modular χ -lattices, Yokohama Math. J. 31 (1983), 13-20. (1983) Zbl0532.06002MR0734154
  10. Snášel V., λ -lattices, Math. Bohemica 122 (1997), 367-372. (1997) MR1600648

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