Another generalization of Higman's well quasi order result on Sigma*

Abstract We present another generalization of Higman's result ([2]) that the subsequence embedding relationship on a finitely generated free monoid is a well quasi order. Our result is analogous to the generalization of Higman's result given in [1] in the following manner. In the quasi orders we consider, we are given a finite set S of non-null words from Σ ∗ and we demand that x1…xk+1 is less than or equal to x1a1…xkakxk+1 for any words x 1 ,…,x k+1 ϵΣ ∗ and any single word a1…ak ϵ S, where aiϵΣ, 1⩽i⩽k. The least quasi order obtained in this fashion is denoted ⪅s. In contrast to the notion of subword unavoidability used in [1], S is said to be subsequence unavoidable (in Σ ∗ ) if and only if there exists a k0 such that any word longer than k0 has a (nonempty) subsequence of letters (not necessarily contiguous) which form a word in S. We show that ⪅s is a well quasi order on Σ ∗ if and only if S is subsequence unavoidable in Σ ∗ . As an application of this result, we show that the iterated shuffle ([1], [7], [8]) of a finite set S with itself is a regular language if and only if S is subsequence unavoidable.