A notion of closure operator with respect to a functor U is introduced. This allows us to describe a number of mathematical constructions that could not be described by means of the already existing notion of closure operator. Some basic results and examples are provided. Mathematics Subject Classification (1991). 18A22, 18A32, 06A15.

In this talk I will give a overview on the connections between closure operators and choice operators and on related results. An operator on a finite set S is a map defined on the set P(S) of all the subsets of S. A closure operator is an extensive, isotone and idempotent operator. A choice operator c is a contracting operator (c(A) ⊆ A, for every A ⊆ S). Choice operators and their lattices hav...

The least common ancestor on two vertices, denoted , , is a well defined operation in a directed acyclic graph (dag) . We introduce , a natural extension of , for any set of vertices. Given such a set , one can iterate in order to obtain an increasing set sequence. being finite, this sequence has always a limit which defines a closure operator. Two equivalent definitions of this operator are gi...

The distribution function for the first eigenvalue spacing in the Laguerre unitary ensemble of finite size may be expressed in terms of a solution of the fifth Painlevé transcendent. The generating function of a certain discontinuous linear statistic of the Laguerre unitary ensemble can similarly be expressed in terms of a solution of the fifth Painlevé equation. The methodology used to derive ...

We consider the question of taking transitive closures on top of pure relational systems (Sybase and Ingres in this case). We developed three kinds of transitive closure programs , one using a stored procedure to simulate a built-in transitive closure operator, one using the C language embedded with SQL statements to simulate the iterated execution of the transitive closure operation, and one u...

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.

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...