Annihilators and deductive systems in commutative Hilbert algebras
Chajda, Ivan ; Halaš, Radomír ; Jun, Young Bae
Commentationes Mathematicae Universitatis Carolinae, Tome 43 (2002), p. 407-417 / Harvested from Czech Digital Mathematics Library
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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 $\operatorname{Ded} H$ 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 $H$.

Publié le : 2002-01-01
Classification:  03B22,  03G10,  03G25,  06A11 
@article{119331,
     author = {Ivan Chajda and Radom\'\i r Hala\v s and Young Bae Jun},
     title = {Annihilators and deductive systems in commutative Hilbert algebras},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     volume = {43},
     year = {2002},
     pages = {407-417},
     zbl = {1070.03043},
     mrnumber = {1920517},
     language = {en},
     url = {http://dml.mathdoc.fr/item/119331}
}

Chajda, Ivan; Halaš, Radomír; Jun, Young Bae. Annihilators and deductive systems in commutative Hilbert algebras. Commentationes Mathematicae Universitatis Carolinae, Tome 43 (2002) pp. 407-417. http://gdmltest.u-ga.fr/item/119331/

Balbes R.; Dwinger P. Distributive Lattices, University of Missouri Press, 1974. | MR 0373985 | Zbl 0321.06012

Busneag D. A note on deductive systems of a Hilbert algebra, Kobe J. Math. 2 (1985), 29-35. (1985) | MR 0811800 | Zbl 0584.06005

Busneag D. Hilbert algebras of fractions and maximal Hilbert algebras of quotients, Kobe J. Math. 5 (1988), 161-172. (1988) | MR 0990817 | Zbl 0676.06018

Busneag D. Hertz algebras of fractions and maximal Hertz algebras of quotients, Math. Japon. 39 (1993), 461-469. (1993) | MR 1278859

Chajda I. The lattice of deductive systems on Hilbert algebras, Southeast Asian Bull. Math., to appear. | MR 2046584 | Zbl 1010.03054

Chajda I.; Halaš R. Congruences and ideals in Hilbert algebras, Kyungpook Math. J. 39 (1999), 429-432. (1999) | MR 1728116

Chajda I.; Halaš R. Stabilizers of Hilbert algebras, Multiple Valued Logic, to appear.

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) | MR 1690472

Diego A. Sur les algébras de Hilbert, Ed. Hermann, Colléction de Logique Math. Serie A 21 (1966), 1-52.

Hong S.M.; Jun Y.B. On a special class of Hilbert algebras, Algebra Colloq. 3:3 (1996), 285-288. (1996) | MR 1412660 | Zbl 0857.03040

Hong S.M.; Jun Y.B. On deductive systems of Hilbert algebras, Comm. Korean Math. Soc. 11:3 (1996), 595-600. (1996) | MR 1432264 | Zbl 0946.03079

Jun Y.B. Deductive systems of Hilbert algebras, Math. Japon. 43 (1996), 51-54. (1996) | MR 1373981 | Zbl 0946.03079

Jun Y.B. Commutative Hilbert algebras, Soochow J. Math. 22:4 (1996), 477-484. (1996) | MR 1426553 | Zbl 0864.03042

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)