Some results in local Hilbert algebras

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Abstract In this paper, we study local Hilbert algebras and give some theorems that characterize local Hilbert algebras. Also, we prove that, (i) H is a local Hilbert algebra if and only if the set of all dense elements of H is H ∖ {0}. (ii) H is a local Hilbert algebra if and only if the set of all regular elements of H is a local Boolean algebra. In a local Hilbert algebra H with supremum, we prove that: (i) If F is a Boolean filter of H, then F is a prime filter. (ii) F is a Boolean filter iff F is a maximal filter iff F = H ∖ {0}.