It is widely known that closure operators on finite sets can be represented by sets of implications (also known as inclusion dependencies) as well as by formal contexts. In this article, we consider these two representation types, as well as generalizations of them: extended implication sets and context families. We discuss the mutual succinctness of these four representations and the tractabil...

(Received) We study the Weyl closure Cl(L) = K(x)h@iL \ D for an operator L of the rst Weyl algebra D = Khx;@i. We give an algorithm to compute Cl(L) and we describe its initial ideal under the order ltration. Our main application is an algorithm for constructing a Jordan-HH older series for a holonomic D-module and a formula for its length. Using the closure, we also reproduce a result of Strr...

A notion of connectedness with respect to a closure operator C and a class of monomorphisms N is introduced in an arbitrary category X . It is shown that under appropriate hypotheses, most classical results about topological connectedness can be generalized to this setting. Examples that illustrate this new concept are provided. Mathematics Subject Classification (1991). 18D35, 06A15, 54D05.

In many database applications, designers can easily provide at least some information about the relative importance of the information to be stored and manipulated. While of potentially high value, the ordering information is typically only partial. Here we address the issue of updates in such partially ordered situations, which we call epistemically stratiied databases. In the current database...

We propose a generalization of continuous lattices and domains through the concept of enriched closure space, defined as a closure space equipped with a preclosure operator satisfying some compatibility conditions. In this framework we are able to define a notion of waybelow relation; an appropriate definition of continuity then naturally follows. Characterizations of continuity of the enriched...

We formulate an infinitary proof system for transitive closure logic, which is the logic obtained by augmenting first-order logics with a transitive closure operator. Our system is an infinite descent-style counterpart to the existing (explicit induction) proof system for the logic. We show that, as for similar systems for first order logic with inductive definitions, our infinitary system is c...

Many problems in data mining can be viewed as a special case of the problem of enumerating the closed elements of an independence system with respect to some specific closure operator. Motivated by real-world applications, e.g., in track mining, we consider a generalization of this problem to strongly accessible set systems and arbitrary closure operators. For this more general problem setting,...