Cobweb Posets and KoDAG Digraphs are Representing Natural Join of Relations, their diBigraphs and the Corresponding Adjacency Matrices
Abstract
Natural join of di-bigraphs that is directed biparted graphs and their corresponding adjacency matrices is defined and then applied to investigate the so called cobweb posets and their Hasse digraphs called KoDAGs. KoDAGs are special orderable directed acyclic graphs which are cover relation digraphs of cobweb posets introduced by the author few years ago. KoDAGs appear to be distinguished family of Ferrers digraphs which are natural join of a corresponding ordering chain of one direction directed cliques called di-bicliques. These digraphs serve to represent faithfully corresponding relations of arbitrary arity so that all relations of arbitrary arity are their subrelations. Being this chain -way complete if compared with kompletne Kuratowski bipartite graphs their DAG denotation is accompanied with the letter K in front of descriptive abbreviation oDAG. The way to join bipartite digraphs of binary into multi-ary relations is the natural join operation either on relations or their digraph representatives. This natural join operation is denoted here by symbol deliberately referring to the direct sum of adjacency matrices as it becomes the case for disjoint di-bigraphs.