Upper bounds for the length of non-associative algebras

We obtain a sharp upper bound for the length of arbitrary non-associative algebra and present an example demonstrating the sharpness of our bound. To show this we introduce a new method of characteristic sequences based on linear algebra technique. This method provides an efficient tool for computing the length function in non-associative case. Then we apply the introduced method to obtain an upper bound for the length of an arbitrary locally complex algebra. We also show that the obtained bound is sharp. In the last case the length is bounded in terms of Fibonacci sequence

Generators of matrix incidence algebras

Let n E Z+ and let K be a field. Let ~ be a partial order on {1, 2,..., n}. Let An(:::;) be the matrix incidence algebra consisting of those n x n matrices A = (ai,j) with entries in K, satisfying ai,j 0 whenever i 1:. j. For a subset £ ~ An (::5), a necessary and sufficient condition that the algebra generated by £ u {I} is An(::5) is that (i) for every 1:::; i, j:::; n with i =1 = j, there exists A E £ such that ai,i =1 = aj,j and (ii) for every i ~ j with j covering i, there exists B E span £ such that bi,j =1 = 0 and bi,i = bj,j. If the characteristic of K is zero or> n, the algebra An (=) is singly generated and, if::5 is not equality, An C::;) has two generators. 1. PRELIMINARIES Let (8, ~) be a locally finite partially ordered set. Here, local finiteness means that every interval [x, y] = {u E 8: x ~ u ~ y} is finite. Let K be a field. The incidence algebra A ( 8) of 8 over K is the set of all functions I: 8 x S-+