On natural join of posets properties and first applications.

In early beginnings of the past century Felix Hausdorff introduced the concept of a partially ordered set thus extending Richard Dedekind lattice theory which began in the early 1890s. Then the subject lay more or less dormant until Garrett Birkhoff, Oystein Ore and others picked it up in the 1930s. Since then, many noted mathematicians have contributed to the subject, including Garrett Birkhoff, Richard Dedekind, Israel Gelfand, George Grtzer, Aleksandr Kurosh, Anatoly Malcev, Oystein Ore, Gian-Carlo Rota, Alfred Tarski and Johnny von Neumann (see Garrett Birkhoff [1] for necessary relevant definitions including operations on posets). Now poset and lattice theories are powerful theories of accelerated growth inseparably tied with graph theory. Here in addition to the three operations on posets from [1] one introduces the natural join P⊕→ Q of posets 〈P,≤P 〉 and 〈Q,≤Q〉 which should satisfy the following Natural Join Conformity Condition: partial orders ≤P and ≤Q must be equivalent on corresponding nonempty R ⊆ P and R′ ⊆ Q isomorphic sub-posets R and R′ which we then shall consider identical by convention. In particular cases P⊕→ Q may be expressed via ordinal sum ⊕ accompanied with projection out operation as it is the case with sequences of antichains forming all together r-ary complete (≡ universal) relations via cardinal sum of these trivial posets. For other, earlier definitions of natural join of graded posets in terms of natural join of zeta matrices see [2,3,4,5,6] or for example Definitions 12 or 2 in this paper. The present definition of natural join P⊕→ Q of posets offers many conveniences. For example one arrives at a very simple proof of the Möbius function formula for cobweb posets. The main aims of this article are to present at first the authors update applications of natural join of posets and/or their cover relation Hasse digraphs and then one summarizes more systematically the general properties of natural join of posets with posing some questions arising on the way. Thus in this note apart from revealing some general properties of natural join of posets we also quote the authors combinatorial interpretations of cobweb posets and explicit formulas for the zeta matrix [2-23] as well as the inverse of zeta matrix for any graded posets [2,3] using Knuth notation [24] while following [3,2]. These formulas use the supplemented by sub-matrix factors corresponding formulas for cobweb posets and their Hasse digraphs named KoDAGs, which are interpreted also as chains of binary complete (or universal) relations joined by the natural join operation see [2,3,4,5,6,12]. Such cobweb posets and equivalently their Hasse digraphs named KoDAGs are also encoded by discrete hyper-boxes [12] and the natural join operation of such discrete hyper-boxes is just Cartesian product of them accompanied with projection out of sine qua non common faces [2]. All graded posets with no mute vertices in their Hasse diagrams [2] (i.e. no vertex has in-degree or out-degree equal zero) are natural