On weak implication algebra

In this paper, we introduce the notion of implication filter as a generalization of Boolean filter of first kind in Hilbert algebra and study it in detail. Also, we prove that F is an implication filter of a Hilbert algebra H if and only if every implicative filter of quotient algebra H/F is an implication filter. Finally, we generalized Boolean algebra and introduced a weak implication algebra, we prove that F is an implication filter of a Hilbert algebra H if and only if the quotient algebra H/F is a weak implication algebra. By suitable diagrams, we summarize the results of this paper and the previous results in these fields.