Green’s Symmetries in Finite Digraphs

Abstract: The semigroup D_V of digraphs on a set V of n labeled vertices is defined. It is shown that D_V is faithfully represented by the semigroup B_n of n ´ n Boolean matrices and that the Green’s L, R, H, and D equivalence classifications of digraphs in D_V follow directly from the Green’s classifications already established for B_n. The new results found from this are: (i) L, R, and H equivalent digraphs contain sets of vertices with identical neighborhoods which remain invariant under certain one-sided semigroup multiplications that transform one digraph into another within the same equivalence class, i.e., these digraphs exhibit Green’s isoneighborhood symmetries; and (ii) D equivalent digraphs are characterized by isomorphic inclusion lattices that are generated by their out-neighborhoods and which are preserved under certain two-sided semigroup multiplications that transform digraphs within the same D equivalence class, i.e., these digraphs are characterized by Green’s isolattice symmetries. As a simple illustrative example, the Green’s classification of all digraphs on two vertices is presented and the associated Green’s symmetries are identified.