Unions of rank/trace complete preorders

Abstract

A zero pattern algebra is a matrix algebra determined by a pattern of zeros corresponding to a preorder (i.e., a reflexive transitive digraph). The issue studied in [6] is: under what conditions is a matrix in a zero pattern algebra A a sum of (rank one) idempotents in A or a logarithmic residue in A? Here logarithmic residues are contour integrals of logarithmic derivatives of analytic A-valued functions. As has been established in [6], there is a necessary condition involving certain rank/trace requirements. Algebras for which this necessary condition is also sufficient are said to be rank/trace complete. Several classes of rank/trace complete algebras are identified in [6]. Also operations are described there for producing rank/trace complete algebras out of given ones. A basic operation of that type is taking a disjoint union. In the present article this simple operation is generalized to a more involved one which preserves rank/trace completeness too. The new operation turns out to be a useful tool for establishing rank/trace completeness in situations that up to now could not be handled. One of the positive results is concerned with bouquets, a special type of rooted trees. It provides a partial answer to an issue raised in [7].