In [O3], we introduced the notion of presemistar operation as a generalization of the notion of semistar operation and gave some characterizations of presemistar operations. In this paper we shall give some new characterizations of presemistar operations. We also introduced the notion of an s.e.a.b. presemistar operation in [O3]. One of the main results of this paper is to show that there exists an example of a proper presemistar operation which is e.a.b. but is not s.e.a.b.. We also give two methods of construction which will provide many new examples of presemistar operations.