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Keywords:
(commutative) Hilbert algebra; deductive system (generated by a set); annihilator
(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 $H$.
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