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Hilbert algebras are dual isomorphic to positive implicative BCK-algebras. (English) Zbl 0930.06017

Summary: We show that the class \({\mathbf H}\) of Hilbert algebras are dual isomorphic to the class \({\mathbf p}{\mathbf i}{\mathbf B}\) of positive implicative BCK-algebras, that is, there exists a map \(\xi: {\mathbf P}_{{\mathbf H}}\to {\mathbf P}_{\mathbf B}\) and a map \(\eta:{\mathbf P}_{\mathbf B}\to{\mathbf P}_{{\mathbf H}}\) such that
(1) if \(p=q\in{\mathbf T}{\mathbf h}({\mathbf H})\) then \(\xi(p)= \xi(q)\in{\mathbf T}{\mathbf h}({\mathbf p}{\mathbf i}{\mathbf B})\), and conversely
(2) if \(r= s\in{\mathbf T}{\mathbf h}({\mathbf p}{\mathbf i}{\mathbf B})\) then \(\eta(r)= \eta(s)\in{\mathbf T}{\mathbf h}({\mathbf H})\),
where \({\mathbf P}_{{\mathbf H}}\) (or \({\mathbf P}_{{\mathbf B}})\) is set of all polynomials of \({\mathbf H}\) \(({\mathbf p}{\mathbf i}{\mathbf B})\). Moreover, we have \(\eta\circ\xi= \text{id}_{{\mathbf P}_{{\mathbf H}}}\) and \(\xi\circ\eta= \text{id}_{{\mathbf P}_{{\mathbf B}}}\).

MSC:

06F35 BCK-algebras, BCI-algebras
03G25 Other algebras related to logic
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