The First Ascent into the Incidence Algebra of the Fibonacci Cobweb Poset

The explicite formulas for Möbius function and some other important elements of the incidence algebra are delivered. For that to do one uses Kwaśniewski’s construction of his Fibonacci cobweb poset in the plane grid coordinate system. 1 Fibonacci cobweb poset The Fibonacci cobweb poset P has been invented by A.K.Kwaśniewski in [1, 2, 3] for the purpose of finding combinatorial interpretation of fibonomial coefficients and eventually their reccurence relation. At first the partially ordered set P (Fibonacci cobweb poset) was there defined via Hasse diagram as follows: P looks like famous rabbits growth tree but it has the specific cobweb in addition, i.e. it consists of levels labeled by Fibonacci numbers (the n-th level consist of Fn elements). Every element of n-th level (n ≥ 0) is in partial order relation with every element of the (n+1)-th level but it’s not with any element from the level in which he lies (n-th level) except from it. In [1] A. K. Kwaśniewski defined cobweb poset P as infinite labeled digraph oriented upwards as follows: Let us label vertices of P by pairs of coordinates: 〈i, j〉 ∈ N0 ×N0, where the second coordinate is the number of level in which the element of P lies (here it is the j-th level) and the first one is the number of this element in his level (from left to the right), here i.