On a factorisation of free monoids
M. P. Schützenberger · Proceedings of the American Mathematical Society · 1965
A property is given which relates two results of Spitzer [6]; it also relates two results of Chen, Fox and Lyndon [1]; the same remark applies to work of Meyer-Wunderli [5] and M. Hall [3] and to its generalisation by Lazard [4]. These connections are indicated more fully below. In what follows, F is the free monoid generated by a fixed set X and F+ denotes the set of all words of positive length of F. If the words f and f' of F belong to a submonoid F' of F, the words ff' an(1 f'f are said to be F'-conjugate. We consider the following conditions, I, I' and II, on a family { Y,: j E J} of subsets of F+ indexed by a totally ordered set J. (I) (resp. (I')). Each fCF+ has at most (resp. at least) one representation in the form f =ff2 ... fn, n > 0, where each fi C Yi, and jl >Jj2>. . * >n. (II) Each F-conjugate class C has nonempty intersection with the submonoid Fj generated by Yj for exactly one jC-J; further, CnF, is an Fj-conjugate class.