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- W1978856360 abstract "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 of a free semigroup S is free if and only if' I. For any a C T, s CE S, if as E T and sa E T, then s E T. 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 S. 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 S 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 x E T. 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 [1]), 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)." @default.
- W1978856360 created "2016-06-24" @default.
- W1978856360 creator A5016875060 @default.
- W1978856360 date "1962-03-01" @default.
- W1978856360 modified "2023-09-25" @default.
- W1978856360 title "On subsemigroups of free semigroups" @default.
- W1978856360 cites W2083662759 @default.
- W1978856360 doi "https://doi.org/10.1090/s0002-9939-1962-0139680-3" @default.
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