Cobweb posets as noncommutative prefabs
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Abstract
We present here a new class of graded infinite posets with minimal element being inspired by the famous Fibonacci tree. These partially ordered sets were introduced earlier by the author as the so called cobweb posets. These cobweb posets are the basis of a class of prefab schemes constructed here below. Prefab schemes are known as close relatives of exponential structures. The prefab schemes to follow constitute a wide range of substantially new prefab combinatorial schemes because new type schemes are defined here via relaxing commutativity and associativity requirements imposed on the composition in prefabs by the fathers of this fertile concept. Characteristic graded sub-posets of cobweb posets are primes of prefab schemes proposed. Apart from the construction also some very first basic properties of cobweb prefabs are disclosed. In addition an another new type prefab example with single valued commutative and associative composition is provided. We start by recalling [revisited] a cobweb combinatorial interpretation of all classical F −nomial coefficients hence specifically incidence coefficients of reduced incidence algebras of full binomial type also. Natural unanswered questions arising during the investigation are posed.
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