Unique Factorization Theorems for Subalgebras of the Incidence Algebra

Abstract

H. Scheid [4] has found necessary and sufficient conditions on a partially ordered set S(≦) which is a direct sum of a countable number of trees for a certain subalgebra G(+, *) of the incidence algebra F(+, *) to be an integral domain. In this paper we prove that under similar conditions on S, G(+, *) is actually a unique factorization domain or, failing this, that there is a subalgebra H(+, *) of F(+, *) which is a unique factorization domain and contains G. Similar results are then obtained as corollaries in the regular convolution rings of Narkiewicz.