On cobweb posets most relevant codings

One considers here acyclic digraphs named KoDAGs (****) which represent the out-most general chains of di-bi-cliques denoting thus the outmost general chains of binary relations. Because of this fact KoDAGs start to become an outstanding concept of nowadays investigation. We propose here examples of codings of KoDAGs looked upon as infinite hyper-boxes as well as chains of rectangular hyper-boxes in N ∞. Neither of KoDAGs' codings considered here is a poset isomorphism with Π = P, ≤. Nevertheless every example of coding supplies a new view on possible investigation of KoDAGs properties. The codes proposed here down are by now recognized as most relevant codings for practical purposes including visualization. More than that. Employing quite arbitrary sequences F = {nF } n≥0 infinitely many new representations of natural numbers called an F-base or base-F number system representations are introduced. These constitute mixed radix-type numeral systems. F-base non-standard positional numeral systems in which the numerical base varies from position to position have picturesque interpretation due to KoDAGs graphs and their correspondent posets which in turn are endowed on their own with combinatorial interpretation of uniquely assigned to KoDAGs F − nomial coefficients. The base-F number systems are used for KoDAGs' coding and are interpreted as chain coordinatization in KoDAGs pictures as well as systems of infinite number of boxes' sequences of F-varying containers capacity of subsequent boxes. Needless to say how crucial is this base-F number system for KoDAGs-hence-consequently for arbitrary chains of binary relations. New F-based numeral systems are umbral base-F number systems in a sense to be explained in what follows.