Free inverse monoids up to rewriting

In this paper, generalizing the study of free partially commutative inverse monoids [5], for any rewriting system T over an alphabet A, we define the notion of T-compatible inverse A-generated monoids, we show there is a free T-compatible monoid FIM(A,T) generated by A and we provide an explicit construction of this monoid. Then, as examples, free partially commutative inverse monoid and free partially semi-commutative inverse monoids are studied and shown to have effective representations.