# Article

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Keywords:
(commutative) Hilbert algebra; deductive system (generated by a set); annihilator
Summary:
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\$.
References:
[1] Balbes R., Dwinger P.: Distributive Lattices. University of Missouri Press, 1974. MR 0373985 | Zbl 0321.06012
[2] Busneag D.: A note on deductive systems of a Hilbert algebra. Kobe J. Math. 2 (1985), 29-35. MR 0811800 | Zbl 0584.06005
[3] Busneag D.: Hilbert algebras of fractions and maximal Hilbert algebras of quotients. Kobe J. Math. 5 (1988), 161-172. MR 0990817 | Zbl 0676.06018
[4] Busneag D.: Hertz algebras of fractions and maximal Hertz algebras of quotients. Math. Japon. 39 (1993), 461-469. MR 1278859
[5] Chajda I.: The lattice of deductive systems on Hilbert algebras. Southeast Asian Bull. Math., to appear. MR 2046584 | Zbl 1010.03054
[6] Chajda I., Halaš R.: Congruences and ideals in Hilbert algebras. Kyungpook Math. J. 39 (1999), 429-432. MR 1728116
[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. MR 1690472
[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. MR 1412660 | Zbl 0857.03040
[11] Hong S.M., Jun Y.B.: On deductive systems of Hilbert algebras. Comm. Korean Math. Soc. 11:3 (1996), 595-600. MR 1432264 | Zbl 0946.03079
[12] Jun Y.B.: Deductive systems of Hilbert algebras. Math. Japon. 43 (1996), 51-54. MR 1373981 | Zbl 0946.03079
[13] Jun Y.B.: Commutative Hilbert algebras. Soochow J. Math. 22:4 (1996), 477-484. MR 1426553 | Zbl 0864.03042
[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.

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