## On deductive systems of Hilbert algebras.(English)Zbl 0946.03079

Summary: We give a characterization of a deductive system. We introduce the concept of maximal deductive systems and show that every bounded Hilbert algebra with at least two elements contains at least one maximal deductive system. Moreover, we introduce the notions of radical and semisimplicity in a Hilbert algebra and prove that if $$H$$ is a bounded Hilbert algebra in which every element is an involution, then $$H$$ is semisimple.

### MSC:

 03G25 Other algebras related to logic 06D99 Distributive lattices