We here consider transition systems of Elementary Net Systems with Inhibitor Arcs. There are basically two different types of non-interleaving semantics of such Petri nets, the a-posteriori and a-priori semantics. The synthesis problem for Elementary Net Systems with Inhibitor Arcs executed under the a-priori semantics (ENI) was solved in [17 ].The aim of this paper is to completely characterise transition systems which can be generated by Elementary Net Systems with Inhibitor Arcs executed under the a- posteriori semantics (ENIapost).This is achieved by adapting the notion of a step transition system, i.e. one in which arcs are labelled by sets of events executed concurrently. In developing the model, we follow the standard approach in which the relationship between nets and their transition systems is established via the notion of a region. We define, and show consistency of, two behaviour preserving translations between nets and transition systems. We then compare transition systems which are generated by ENIapost and ENI-systems (called respectively TSENIapost and TSENI transition systems).