Semirings of cyclic types. (Q950841)

The authors study the semirings with a cyclic structure. Let \(A\) be an Abelian group and \(\sigma\colon A\to S(\Omega)\) be a representation of \(A\) by permutations of the set \(\Omega\), \(\Omega_i\), \(i=1,\dots,m\), be the orbits of the group \(\sigma(A)\) on \(\Omega\), \(\sigma_i\colon A\to S(\Omega)\) being the homomorphism induced by \(\sigma\) on the set of orbits, \(K_i=\text{Ker\,}\sigma_i\). Denote by \(H(A)\) the set of all subgroups of \(A\) having finite index. Then \(\mathbf H=H(A)\) is a semigroup with respect to the set-intersection and \(\mathbb{N}\mathbf H\) is structured as a semiring over \(\mathbb{N}\). The polynomial \(Z_\sigma=\sum K_i\), for \(i=1,\dots,m\), is called the cyclic type of the given representation. The authors give the cyclic type of the product of two representation, \(\sigma\times\tau\colon A\to S(\Omega\times\Delta)\). They also establish some properties of the semirings defined in the same manner by using a semiring \(R\) instead of the semiring \(\mathbb{N}\) of natural numbers in connection with their invertible elements, zero divisors and Jacobson radical (when \(R\) is a ring).