Syntactic and Rees Indices of Subsemigroups

We define two different notions of index for subsemigroups of semigroups: the (right) syntactic index and the Rees index. We investigate the relationships between them and with the group index. In particular, we show that the syntactic index is a generalisation of both the group index and the Rees index. We use this fact to prove further similarities between the group index and the Rees index. It turns out, however, that very few of the nice properties of the group index are inherited by the syntactic index