ErdH{o}s [On Sch"utte problem, Math. Gaz. 47 (1963)] proved that every tournament on $n$ vertices has a directed dominating set of at most $log (n+1)$ vertices, where $log$ is the logarithm to base $2$. He also showed that there is a tournament on $n$ vertices with no directed domination set of cardinality less than $log n - 2 log log n + 1$. This notion of directed domination number has been g...

This paper studies the complexity of query evaluation for databases whose relations are partially ordered; the problem commonly arises when combining ordered data from multiple sources. We focus on queries in a useful fragment of SQL, namely positive relational algebra with aggregates, whose bag semantics we extend to the partially ordered setting. Our semantics leads to the study of two main c...

Consider edge colorings of directed graphs where edges of the form v1v2 and v2v3 must have different colors. Here, v1 ≠ v2 , v2 ≠ v3 but v1 = v3 is possible. It is known that this coloring induces a vertex coloring by sets of edge colors, in which edge v1v2 in the graph implies that the set color of v1 contains an element not in the set color of v2; conversely, each such set coloring of vertice...

This paper presents the extension of results on reasoning with totally ordered belief bases to the partially ordered case. The idea is to reason from logical bases equipped with a partial order expressing relative certainty and to construct a partially ordered deductive closure. The difficult point lies in the fact that equivalent definitions in the totally ordered case are no longer equivalent...

In the late 1930's Garrett Birkhoff [3] pioneered the theory of distributive lattices. An important component in this theory is the concept of exponentiation of lattices [4]: for a lattice L and a partially ordered set P let L denote the set of all order-preserving maps of P to L partially ordered b y / ^ g if and only if/(;c) ^ g(x) for each x e P (see Figure 1). Indeed, If is a lattice. This ...

We reconstruct orthoalgebras from their partially ordered set of Boolean subalgebras, and characterize partially ordered sets of this form, using a new notion of a direction. For Boolean algebras, this reconstruction is functorial, and nearly an equivalence.