On minimal elements of upward-closed sets

Minimal closed sets and maximal closed sets

Some properties of minimal open sets and maximal open sets are studied in [1, 2]. In this paper, we define dual concepts of them, namely, maximal closed set and minimal closed set. These four types of subsets appear in finite spaces, for example. More generally, minimal open sets and maximal closed sets appear in locally finite spaces such as the digital line. Minimal closed sets and maximal op...

Symboli Representation of Upward-Closed Sets

Symbolic Representation of Upward-Closed Sets

The reachability problem for a wide class of in nite-state systems is decidable when the initial and the nal set of con gurations are given as upwards closed sets. Traditional symbolic model checking methods su er from the state explosion problem when applied to this class of veri cation problems. We provide new data structures and algorithms for an eÆcient manipulation of upwards closed sets. ...

On the existence of minimal elements in partially ordered sets

In this work, we give a characterization of the existence of minimal elements in partially ordered sets in terms of fixed point type statement. This characterization shows that the assumptions in Caristi’s fixed point theorem can, a priori, be weakened. Finally, we give a negative answer to Kirk’s problem on an extension of Caristi’s result.

Some Generalizations of Locally Closed Sets

Arenas et al. [1] introduced the notion of lambda-closed sets as a generalization of locally closed sets. In this paper, we introduce the notions of lambda-locally closed sets, Lambda_lambda-closed sets and lambda_g-closed sets and obtain some decompositions of closed sets and continuity in topological spaces.