Rees semigroups of digraphs for classification of data

Recent research has motivated the investigation of the weights of ideals in semiring constructions based on semigroups. The present paper introduces Rees semigroups of directed graphs. This new construction is a common generalization of Rees matrix semigroups and incidence semigroups of digraphs. For each finite subsemigroup S of the Rees semigroup of a digraph and for every zero-divisor-free idempotent semiring F with identity element, our main theorem describes all ideals J in the semigroup semiring F0[S] such that J has the largest possible weight.