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On the Glivenko-Frink theorem for Hilbert algebras. (English) Zbl 1199.03062
The author gives a Glivenko-Frink-type theorem for Hilbert algebras. Such theorems say that for a pseudocomplemented distributive lattice \((L,\wedge,\vee,^*,0)\), the set of its regular elements \(R(L) : = \{ x\in L\mid x^{**}=x\}=\{x^* \mid x\in L\}\) can be transformed into a Boolean algebra. The paper under review proves that in any Hilbert algebra \(A\) that is not necessarily bounded, each principal filter \([a,1]\) is a bounded Hilbert algebra. In addition, for each \(a\in A,\) the mapping \(x \mapsto (x\to a)\to a=:x^{aa}\) is a surjective homomorphism of a special type and this theorem entails a group of Glivenko-Frink-type theorems.

03G25 Other algebras related to logic
06D15 Pseudocomplemented lattices