Genus characters and the arithmetic of commutative monoids (Q1360717)

An arithmetic monoid \(M\) is defined as a subsemigroup of the multiplicative semigroup \(Z(\Delta)\) of rational integers prime to a given integer \(\Delta\) satisfying the following condition: for any \(c\in Z(\Delta)\) there is an integer \(m\geq1\) with \(c^m\in M\). The authors define and study ideal theory of such monoids defining two class groups \(CL(M)\) and \(CL'(M)\), analogous to class groups arising in commutative algebra. If \(\chi\) is a character of such class group, then one can associate with it a function \(L(s,\chi)\) regular in the half-plane Re \(s>1\), which in case \(M=Z(\Delta)\) coincides with the usual Dirichlet \(L\)-function. Another example arises when \(M=M_K\) is the set of rational integers which are norms of non-zero elements of a fixed quadratic field \(K\), in which case the group \(CL(M_K)'\) is isomorphic to the group of genera of \(K\). In the last part of the paper the case \(M=\bigcap M_{K_j}\) is considered, where \(K_1,\dots,K_k\) are distinct quadratic fields. It is shown a.o. that if the discriminants \(d(K_j)\) are pairwise coprime then \(CL(M)\) is an elementary 2-group of rank \(\Sigma\omega (d(K_j))-\kappa\).