OF DIVISORS OF A GENERALIZED INTEGER ŠTEFAN PORUBSKÝ

Dedicated to the memory of Professor Ivan Korec Abstract. The notion of a regular system of divisors of a number underlying a Narkiewicz's idea of the generalization of the familiar Dirichlet and unitary convolution of arithmetical functions is used to extend some combinatorial results about systems of subsets of divisors of a generalized integer. In 1963 W. Narkiewicz [Narkl963] introduced a class of binary operations on arithmetical functions generalizing the known Dirichlet and unitary co nvolution. The generalization was based on the idea to associate with every positive integer n not the whole set of all divisors or the set of all unitary divisors of n, resp., but only a subset of the set of all divisors of n which fulfils certain properties. In the original setting these properties were then determined by the properties imposed on the resulting convolution. In what follows we shall not concern more about the original convolution backgro und. The idea is capable of further extension to the so called arithmetical semigroups. Here (cf. [Knop75]), an a r i thmet ica l semigroup is a com mutative semigroup (G,.) with identity element 1 = 1Q and with at most countable set P of generators (called primes of G) such that every elemen