Geodesic
Annihilators and deductive systems in commutative Hilbert algebras
Commentationes Mathematicae Universitatis Carolinae, Tome 43 (2002) no. 3, pp. 407-417 Cet article a éte moissonné depuis la source 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 $H$.
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 $H$.
Classification : 03B22, 03G10, 03G25, 06A11
Keywords: (commutative) Hilbert algebra; deductive system (generated by a set); annihilator 
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     title = {Annihilators and deductive systems in commutative {Hilbert} algebras},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {407--417},
     year = {2002},
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     zbl = {1070.03043},
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Chajda, I.; Halaš, R.; Jun, Y. B. Annihilators and deductive systems in commutative Hilbert algebras. Commentationes Mathematicae Universitatis Carolinae, Tome 43 (2002) no. 3, pp. 407-417. http://geodesic.mathdoc.fr/item/CMUC_2002_43_3_a1/