Intermediate logics with the same disjunctionless fragment as intuitionistic logic

Given an intermediate prepositional logic L, denote by L^–dits disjuctionless fragment. We introduce an infinite sequence {J_n}_n1 of propositional formulas, and prove:(1)For anyL: L^–d=I^–d (I=intuitionistic logic) if and only if J_n Lfor every n 1.Since it turns out that L{J_n}n1 = Ø for any L having the disjunction property, we obtain as a corollary that L^–d = I^–d for every Lwith d.p. (cf. open problem 7.19 of [5]). Algebraic semantic is used in the proof of the if part of (1). In the last section of the paper we provide a characterization in Kripke's semantic for the logics J_n=I+ +J_n (n 1).