# zbMATH — the first resource for mathematics

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.

##### MSC:
 03G25 Other algebras related to logic 06D15 Pseudocomplemented lattices
##### Keywords:
Hilbert algebra; Glivenko-Frink theorem