2009/07/15 by A. K. Kwasniewski, A. K. Kwaśniewski, Kwasniewski, A. K.
Chemistry · Computer Science · Mathematics · #05B20 #05C75 #06A06 #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Mathematics #Molecular spectroscopy and chirality #Topological and Geometric Data Analysis #math.CO #msc:05B20 #msc:05C75 #msc:06A06
paper · pdf · doi:10.48550/arxiv.0907.2595
51 pages, 11 figures, The Internet Gian-Carlo Rota Polish Seminar article http://ii.uwb.edu.pl/akk/sem/sem\_rota.htm
openalex publication_date 2009/07/15 · arxiv created 2009/07/30 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
One introduces here the natural join P \os Q of graded posets < P,≤P > and < Q,≤Q > with correspondingly maximal and minimal sets being identical as expressed by ordinal sum P⊕ Q apart from other definition and due to that one arrives at a simple proof of the Möbius function formula for cobweb posets. We also quote the other authors explicit formulas for the zeta matrix and its inverse for any graded posets with the finite set of minimal elements from earlier works of the author. These formulas are based on the formulas for cobweb posets and their Hasse diagrams or graphs named KoDAGs which are interpreted as chains of binary complete or universal relations joined by the natural join operation. Natural join of two independent sets is therefore the ordinal sum of this trivially ordered posets represented also by directed biclique named dibiclique and correspondingly by their Hasse diagrams or graphs named KoDAGs. Such cobweb posets and equivalently their Hasse diagrams or graphs named KoDAGs are also encoded by discrete hyper-boxes and the natural join operation of such discrete hyper boxes is just cartesian product of them accompanied with projection out of common faces. All graded posets with no mute vertices in their Hasse diagrams which means that no vertex has indegree or outdegree equal zero are natural join of chain of relations and may be at the same time interpreted an n-ary relation, n ∈ N ∪ \∞ \.