Free Commutative Semigroup Research Articles

Is There a Sequence on Four Symbols in Which No Two Adjacent Segments are Permutations of One Another?

It has long been known (see [1–3, 5, 6, 10–12, 15, 16]) that there exist sequences on 3 symbols which contain no 2 identically equal consecutive segments, and sequences on 2 symbols which contain no 3 identically equal consecutive segments. Indeed, Axel Thue obtained these results around 1906. See [6] for a brief account of the contexts of the various independent rediscoveries of these results, and see [7,14] for an account of other properties of these sequences. Let X be a set and let s = x1x2x3 be a sequence on X . Then for i+1 k, s[i+1;k] = xi+1xi+2 xk is a segment of s, and the segments s[i+ 1; j];s[ j + 1;k] are consecutive. The segments s[i+ 1; j] and s[p+ 1;q] are identically equal if k i = q p and xi+1 = xp+1;xi+2 = xp+2; : : : ;xk = xq or, in other words, if s[i+1;k] = s[p+1;q] in X , the free semigroup generated by the set X . An interesting situation arises when we allow the symbols within a segment to commute with each other. It will be convenient to use the following terminology. Given a set X and a sequence s on X , we regard segments of s as elements of X . (Thus the results mentioned above say that there exist sequences on 3 symbols without 2nd powers as segments, and sequences on 2 symbols without 3rd powers.) Now let X denote the free commutative semigroup generated by X , and let α : X 7! X be the natural homomorphism (α(x) = x for x 2 X). If s has k consecutive segments f1; : : : ; fk such that α( f1) = = α( fk), then we say that s has a kth power mod α . In this language, the question of the title is: Does there exist a sequence on four symbols without 2nd powers mod α? It is an easy matter to verify that every sequence on 3 symbols contains 2nd powers mod α , and that every sequence on 2 symbols has 3rd powers mod α . For example, if X = fx;yg, one can show by examining all cases that the longest elements of X which do not contain a 3rd power mod α are xxyyxyyxx;xxyyxyyxy; and a few others. Evdomikov [4] constructed a sequence on 25 symbols without 2nd powers mod α , and conjectured that perhaps 5 symbols would suffice. Justin [8], with a remarkable half-page proof, constructed a sequence on 2 symbols without 5th powers mod α . This sequence is obtained by successive iterations of the transformation x 7! xxxxy and y 7! xyyyy, starting with x. Thus the first few iterations give x;xxxxy;(xxxxy)4xyyyy; [(xxxxy)4xyyyy]4xxxxy(xyyyy)4. Then in 1970 a paper appeared [13] in which P. A. B. Pleasants gave a construction of a sequence on 5 symbols without 2nd powers mod α . Pleasants’ sequence, which extends to infinity in both directions, is constructed by

The free envelope of a finitely generated commutative semigroup

Let S be a commutative semigroup. A quasi-universal free semigroup of S is a free commutative semigroup with identity F together with a homomorphism -q of S into F such that any homomorphism of S into a free commutative semigroup with identity factors through -q; if there is uniqueness in this factorization, we say that (F, -q) is a universal free semigroup of S. If S is finitely generated, there exists a smallest quasi-universal free semigroup of S; we call it the free envelope of S. Its construction and study is the first object of this paper, the second being the application of the free envelope to the study of cancellative and power-cancellative commutative semigroups. We construct the free envelope in the first section. Cancellative and powercancellative semigroups appear in ?2; we prove that a finitely generated commutative semigroup is embeddable into a free commutative semigroup with identity if and only if it has these properties and has either no identity or a trivial group of units; then it is embeddable into its free envelope. The study of this latter embedding gives, conversely, a number of interesting properties of the free envelope in the general situation. The dual of a finitely generated commutative semigroup S may be defined as the semigroup S* of all homomorphisms of S into the additive semigroup of all nonnegative integers, under pointwise addition. Using free envelopes, we prove in ?3 that S***-S* and investigate the relationship between S and S**. This yields in turn a number of results concerning universal free semigroups when they exist. A study of various dimensions completes the section. ?4 deals with embeddings Sc T such that every relation which holds in S can be deduced from the presentation of S in T without using any relation which may hold in T (in which case we say that T kills S). We show that, if S is finitely generated, the (inclusion) homomorphism of S into T can be extended to the free envelope of S; if furthermore T is cancellative, power-cancellative and without identity element, then T contains subsemigroups which kill S and are minimal with that property. This paper has benefited from numerous suggestions, by the members of the Tulane semigroup seminar, especially William R. Nico and A. H. Clifford, and by our referee, which we acknowledge gladly.