This article continues the investigation of matrix constructions motivated by their applications to the design of classification systems. Our main theorems strengthen and generalize previous results by describing all centroid sets for classification systems that can be generated as one-sided ideals with the largest weight in structural matrix semirings. Centroid sets are well known in data mining, where they are used for the design of centroid-based classification systems, as well as for the design of multiple classification systems combining several individual classifiers.