Network right * abundant semigroups

We introduce the class of network right $\ast$-abundant semigroups. These are based on networks that extend the notion of a directed graph. This class properly contains the class of graph inverse semigroups. We investigate the structure of network right $\ast$-abundant semigroups. We show that two network right $\ast$-abundant semigroups are isomorphic if and only if the underlying networks are isomorphic.