  One considers here orderable acyclic digraphs named KoDAGs which represent
the outmost general chains of dibicliques 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^\infty. Neither of KoDAGs codings considered here
is a poset isomorphism with Pi = <P, \leq>. 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 codes for
practical purposes including visualization. More than that. Employing quite
arbitrary sequences F=\{n_F\}_{n\geq 0} infinitely many new representations of
natural numbers called base of F number system representations are introduced.
These constitute mixed radix-type numeral systems. F base nonstandard
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
of 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 of F number system for KoDAGs hence
consequently for arbitrary chains of binary relations. New F based numeral
systems are umbral base of F number systems in a sense to be explained in what
follows.
