Chains, antichains, and fibres
Cutset theory is important in the study of graphs and posets and has been much explored with many important properties of these from the standpoint of applications (e.g., in computer architecture) describable in terms of cusets or cocycles. Fibers, as subsets of posets, meet every maximal antichain and may be expected to have properties somewhat dual or complementary to those of cusets. As much as the natural position of fibers in the theory of posets warrants, until this most interesting paper, exploration of the theory of fibers has been much less extensive than might have been expected. The authors have ectified this situation to a good extent with a set of important results which confirm the existence of duality-type properties (e.g., in theorems 4 and 8), provide interesting observations and questions taking as point of departure a conjecture of Aigner and Andreae, meanwhile demonstrating ways to generalize or recast results obtained for posets to some for graphs (e.g., in theorem 5), significant in their own right. Besides fibers, cones and other better known structures in posets, spirals S(C) consisting of all elements comparable with each element of chains C, appear to be useful and much less well-known creatures having strong properties in many situations (e.g., in theorem 2). This paper should remain a fundamental reference in the area for some time to come.
scientific article; zbMATH DE number 4037890