In this study, we investigate a partial order induced by a nullnorm F on a bounded lattice L, called the F-partial order, which was recently introduced by Awl. We consider the case where L = [0, 1]. First, we define and comprehensively study the set of all incomparable elements with arbitrary but fixed x is an element of (0, 1) according to the F-partial order. We then define and discuss the equivalence relation on the class of nullnorms according to the partial orders that they generate. By defining this relation, the equivalence relations defined on the class of t-norms and t-conorms are extended to a more general form. Finally, we provide an answer to an open problem regarding the relation between the lattice order and the F-partial order on [0, 1]. (C) 2017 Elsevier B.V. All rights reserved.