About primitive systems of natural numbers

The set structure of primitive systems of natural numbers is described, and the main properties of such systems are installed. An algorithm for enumerating primitive systems of numbers not exceeding a given number $m$ is constructed using the concepts of deadlockness and $k$-minimalities of primitive systems. Also, some algorithms are offered for determining the primitiveness index of a finite directed graph by means of depth-first search and the exponentiation of the vertex adjacency matrix. Computational complexity of the algorithms is estimated.