On the Glivenko-Frink theorem for Hilbert algebras

The Glivenko-Frink theorem for pseudocomplemented distributive lattices and, more generally, for pseudocomplemented meet semilatices, states that the set of regular elements can be made into a Boolean algebra which is a homomorphic image of the original algebra. Busneag has extended this theorem to bounded Hilbert algebras. In the present paper we work with a Hilbert algebra A which is not supposed to be bounded and prove that each principal order filter [a, 1] is a bounded Hilbert algebra (Theorem 1) whose regular elements form a Boolean algebra which is a homorphic image of [a, 1] (Theorem 2). Then (Theorem 3) for each element a of A we construct a Boolean-like algebra on A and a surjective homomorphism of type (2,2,1,0,0) from this algebra to the Boolean algebra obtained in Theorem 2.