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Stabilizers in Hilbert algebras. (English) Zbl 1024.03065
The concept of Hilbert algebras was introduced in the early fifties by L. Henkin and T. Skolem for investigations in intuitionistic and other nonclassical logics. The main goal of this paper is to describe explicitly the pseudocomplements and relative pseudocomplements in the algebraic lattice of all deductive systems of a Hilbert algebra. For this, the authors introduce the concept of a stabilizer and a relative stabilizer of a given subset of a Hilbert algebra \(\mathcal H\). They prove that every stabilizer of a deductive system \(\mathcal C\) of \(\mathcal H\) is also a deductive system which is a pseudocomplement \(\mathcal C\) in the lattice \(\operatorname {Ded}\mathcal H\) of all deductive systems of \(\mathcal H\). Moreover, every relative stabilizer of \({\mathcal C}\in \operatorname {Ded} {\mathcal H}\) w.r.t. \(B\in \operatorname {Ded} {\mathcal H}\) is a relative pseudocomplement of \({\mathcal C}\) w.r.t. \(B\) in \(\operatorname {Ded}{\mathcal H}\).

03G25 Other algebras related to logic
06B10 Lattice ideals, congruence relations