On Stjbsemigroups of Free Semigroups

1. In a recent paper [4], Sevrin has given necessary and sufficient conditions for a subsemigroup of the free product of a free group and a free semigroup to be of the same form; in particular he deduces that a subsemigroup T oí a free semigroup 5 is free if and only if1 1. For any aET, sES, if as ET and saET, then sET. Now whereas the property of being a free subsemigroup of S is absolute, i.e. it does not depend on the way the subsemigroup is embedded in S, I expresses a condition on T relative to 5. Our object here is to note a criterion similar to I, but which does not refer explicitly to S. Namely, a subsemigroup T of a free semigroup 5 is free if and only if II. For any a, a', b, b'ET, if ab' = ba', then a = bx or b = ax for some xET. It is not difficult to establish the equivalence of I and II (cf. Theorem 2 below), but since II, with a supplementary condition has been used to characterize free semigroups (cf. Dubreil-Jacotin [3], Clifford [l]), it may be of interest to derive the condition in a somewhat wider context. We shall in fact give a characterization of semigroups which can be expressed as the free product of a group and a free semigroup (Theorem 1) ; this will include the case considered by Sevrin. From this result it is easy to obtain the above conditions for a subsemigroup of a free semigroup to be again free (Theorem 2). As a second application we determine the structure of the semigroup of homogeneous elements of a free associative algebra (Theorem 3).