Annihilators and deductive systems in commutative Hilbert algebras

Ivan Chajda; Radomír Halaš; Young Bae Jun

Commentationes Mathematicae Universitatis Carolinae (2002)

  • Volume: 43, Issue: 3, page 407-417
  • ISSN: 0010-2628

Abstract

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The properties of deductive systems in Hilbert algebras are treated. If a Hilbert algebra H considered as an ordered set is an upper semilattice then prime deductive systems coincide with meet-irreducible elements of the lattice Ded H of all deductive systems on H and every maximal deductive system is prime. Complements and relative complements of Ded H are characterized as the so called annihilators in H .

How to cite

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Chajda, Ivan, Halaš, Radomír, and Jun, Young Bae. "Annihilators and deductive systems in commutative Hilbert algebras." Commentationes Mathematicae Universitatis Carolinae 43.3 (2002): 407-417. <http://eudml.org/doc/249013>.

@article{Chajda2002,
abstract = {The properties of deductive systems in Hilbert algebras are treated. If a Hilbert algebra $H$ considered as an ordered set is an upper semilattice then prime deductive systems coincide with meet-irreducible elements of the lattice $\operatorname\{Ded\} H$ of all deductive systems on $H$ and every maximal deductive system is prime. Complements and relative complements of $\operatorname\{Ded\} H$ are characterized as the so called annihilators in $H$.},
author = {Chajda, Ivan, Halaš, Radomír, Jun, Young Bae},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {(commutative) Hilbert algebra; deductive system (generated by a set); annihilator; Hilbert algebra; deductive system; annihilator},
language = {eng},
number = {3},
pages = {407-417},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Annihilators and deductive systems in commutative Hilbert algebras},
url = {http://eudml.org/doc/249013},
volume = {43},
year = {2002},
}

TY - JOUR
AU - Chajda, Ivan
AU - Halaš, Radomír
AU - Jun, Young Bae
TI - Annihilators and deductive systems in commutative Hilbert algebras
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2002
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 43
IS - 3
SP - 407
EP - 417
AB - The properties of deductive systems in Hilbert algebras are treated. If a Hilbert algebra $H$ considered as an ordered set is an upper semilattice then prime deductive systems coincide with meet-irreducible elements of the lattice $\operatorname{Ded} H$ of all deductive systems on $H$ and every maximal deductive system is prime. Complements and relative complements of $\operatorname{Ded} H$ are characterized as the so called annihilators in $H$.
LA - eng
KW - (commutative) Hilbert algebra; deductive system (generated by a set); annihilator; Hilbert algebra; deductive system; annihilator
UR - http://eudml.org/doc/249013
ER -

References

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  2. Busneag D., A note on deductive systems of a Hilbert algebra, Kobe J. Math. 2 (1985), 29-35. (1985) Zbl0584.06005MR0811800
  3. Busneag D., Hilbert algebras of fractions and maximal Hilbert algebras of quotients, Kobe J. Math. 5 (1988), 161-172. (1988) Zbl0676.06018MR0990817
  4. Busneag D., Hertz algebras of fractions and maximal Hertz algebras of quotients, Math. Japon. 39 (1993), 461-469. (1993) MR1278859
  5. Chajda I., The lattice of deductive systems on Hilbert algebras, Southeast Asian Bull. Math., to appear. Zbl1010.03054MR2046584
  6. Chajda I., Halaš R., Congruences and ideals in Hilbert algebras, Kyungpook Math. J. 39 (1999), 429-432. (1999) MR1728116
  7. Chajda I., Halaš R., Stabilizers of Hilbert algebras, Multiple Valued Logic, to appear. 
  8. Chajda I., Halaš R., Zednik J., Filters and annihilators in implication algebras, Acta Univ. Palack. Olomuc, Fac. Rerum Natur. Math. 37 (1998), 141-145. (1998) MR1690472
  9. Diego A., Sur les algébras de Hilbert, Ed. Hermann, Colléction de Logique Math. Serie A 21 (1966), 1-52. 
  10. Hong S.M., Jun Y.B., On a special class of Hilbert algebras, Algebra Colloq. 3:3 (1996), 285-288. (1996) Zbl0857.03040MR1412660
  11. Hong S.M., Jun Y.B., On deductive systems of Hilbert algebras, Comm. Korean Math. Soc. 11:3 (1996), 595-600. (1996) Zbl0946.03079MR1432264
  12. Jun Y.B., Deductive systems of Hilbert algebras, Math. Japon. 43 (1996), 51-54. (1996) Zbl0946.03079MR1373981
  13. Jun Y.B., Commutative Hilbert algebras, Soochow J. Math. 22:4 (1996), 477-484. (1996) Zbl0864.03042MR1426553
  14. Jun Y.B., Nam J.W., Hong S.M., A note on Hilbert algebras, Pusan Kyongnam Math. J. (presently, East Asian Math. J.) 10 (1994), 279-285. (1994) 

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