Cobweb posets as noncommutative prefabs

A class of new type graded infinite posets with minimal element is introduced. These so called cobweb posets proposed recently by the present author constitute a wide range of new noncommutative and nonassociative prefab combinatorial schemes‘ examples with characteristic graded sub-posets as primes. These schemes are defined here via relaxing commutativity and associativity requirements imposed on the composition in prefabs by the fathers of this fertile concept. The construction and the very first basic properties of cobweb prefabs are disclosed. An another new type prefab example with single valued commutative and associative composition is provided. ”En passant” though not by accident we discover new combinatorial interpretation of all classical F − nomial coefficients hence specifically incidence coefficients of reduced incidence algebras of full binomial type are given a new cobweb combinatorial interpretation also.