What is so special about Higman's Lemma?

Question

Is there a motivational example of an application of Higman's Lemma that brings out the true beauty and importance of Higman's Lemma? What is the thing that made so many people care about it? For an example, I was thinking that given a singleton set $\{1\}$ with the linear order $\preceq$ such that basically $1\preceq1$. Now take the free monoid of this singleton and you get an infinite set $\{1,2,3,\cdots\}$ which is the positive natural numbers $\mathbb{N}_+$. If we define the same linear order $\preceq$ on the free monoid $\mathbb{N}_+$, we get $1\preceq2\preceq\cdots$. And therefore the order on the singleton set $\{1\}$ satisfies the free monoid $\mathbb{N}_+$ as well. I am not sure how important this example is but I remember that this is how I came to understand the lemma. I also thought of the huge range of flexibility to this lemma but could not come up with anything to fully encompass the true beauty behind this wonderful lemma. Like finding a generalization/construction in category theory, which is what I am currently working on, and thus opening new doors for discovery.

Answer 1

Not sure this is the answer you expect, but here is an important application of Higman's lemma in formal language theory. Let $A$ be a finite alphabet and let $A^*$ be the free monoid on $A$. The shuffle product of two languages $U$ and $V$ over $A$ is the language \begin{multline*} U \operatorname{ш} V = \{ w \in A^* \mid w = \color{red}{u_1}\color{blue}{v_1} \cdots \color{red}{u_n}\color{blue}{v_n} \text{ for some words $\color{red}{u_1, \ldots, u_n}$} \\ \text{$\color{blue}{v_1, \ldots, v_n}$ of $A^*$ such that $\color{red}{u_1 \cdots u_n} \in U$ and $\color{blue}{v_1 \cdots v_n} \in V$} \}\,. \end{multline*} Note that in this definition, $\color{red}{u_1, \ldots, u_n}$ as well as $\color{blue}{v_1, \dots, v_n}$ can be empty words. The following result is a consequence of Higman's lemma.

Theorem. For each language $L$, the language $L \operatorname{ш} A^*$ is a regular language.

This gives examples of languages that are known to be regular, but for which there is no known finite automaton (or regular expression) representing them, a rather disturbing situation.

This result does not use the full strength of Higman's lemma, only the fact that the subword order on $A^*$ is a well-quasi-order.

Definition. A word $u = \color{red}{a_1a_2 \dotsm a_n}$ (where $a_1, a_2, \dots a_n \in A$) is a subword of a word $v$ if there exist $v_0, v_1, \dots, v_n \in A^*$ such that $v = v_0\color{red}{a_1}v_1\color{red}{a_2}v_2 \dotsm v_{n-1}\color{red}{a_n}v_n$.

By Higman's lemma, the subword order on $A^*$ is a well-quasi-order. Therefore, for each language $L$, the set $F$ of minimal words of $L$ (for the subword ordering) is a finite set $F$ and $L \operatorname{ш} A^* = F \operatorname{ш} A^*$. It is now easy to show that $F \operatorname{ш} A^*$ is a regular language.