Several equivalent conditions for filters are given, and the definition of obstinate filters is
introduced. Obstinate filters, maximal implicative filters and maximal positive implicative filters are
proved equivalent to each other. It is also concluded that ultrafilters are obstinate filters and
obstinate filters are prime filters. An equivalent condition for the obstinate filterLis provided
through implication homomorphism, that is, for arbitrary lattice implication algebraL′,there
always exists an implication homomorphism fromLtoL′,such that this filter is a dual kernel of
this homomorphism.