Annihilators and deductive systems in commutative Hilbert algebras

Abstract

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 $H$.