In this paper we investigate the interplay between isolated suborders and closures. Isolated suborders are a special kind of suborders and can be used to diminish the number of elements of an ordered set by means of a quotient construction. The decisive point is that there are simple formulae establishing relationships between the number of closures in the original ordered set and the quotient thereof induced by isolated suborders. We show how these connections can be used to derive a recursive algorithm for counting closures, provided the ordered set under consideration contains suitable isolated suborders.

Central to algorithmic graph theory are the concepts of acyclicity and strongly connected components of a graph, and the related search algorithms. This article is about combining mathematical precision and concision in the presentation of these concepts. Concise formulations are given for, for example, the reflexive-transitive reduction of an acyclic graph, reachability properties of acyclic graphs and their relation to the fundamental concept of “definiteness”, and the decomposition of paths in a graph via the identification of its strongly connected components and a pathwise homomorphic acyclic subgraph. The relevant properties are established by precise algebraic calculation. The combination of concision and precision is achieved by the use of point-free relation algebra capturing the algebraic properties of paths in graphs, as opposed to the use of pointwise reasoning about paths between nodes in graphs.

Considering initiatives like Industry 4.0 or the Industrial Internet of Things, robots will play an important role in intelligent factories, producing highly customized products with high variability and in small lot sizes. In this setting, complexity of planning and programming such robotic applications grows due to the drastic increase in flexibility, performance and robustness required. In this paper, we propose a tool-supported methodology for the development of control software for dynamically forming multi-functional robot teams. The main challenges for achieving this overall goal are modeling of robot team skills, techniques for automatically deriving process steps from the products’ construction plans, finding allocations of those steps to possible robot teams with compatible skills and calculating collision-free execution schedules with a high degree of parallelization to improve cycle times. The proposed approach integrates process experts and automation experts on all level s. Two case studies will serve as test beds to the developed approach: production of carbon-fiber reinforced polymers and assembly of furniture.