 The constructive implication of positive intuitionistic logic is present in the systems with subminimal negation from Colacito, De Jongh, and Vargas 2017. They also treat negation as a kind of modal operator by considering frames \((W, R, N)\), where \((W, R)\) is a relational frame, \(R\) is a partial order on \(W\), and \(N\) is a unary function on the set of all upwards closed subsets of \(W\) (i.e., the set of all \(X \subseteq W\) such that if \(w \in X\) and \(wRu\), then \(u \in X\)). A model is obtained by adding to a frame a valuation function that is persistent for atomic formulas. Moreover, the following condition is imposed on the function \(N\):