# soft question – Why is the Dyck language/Dyck paths named after von Dyck? The Dyck language is defined as the language of balanced parenthesis expressions on the alphabet consisting of the symbols $$($$ and $$)$$. For example, $$()$$ and $$()(()())$$ are both elements of the Dyck language, but $$())($$ is not. There is an obvious generalisation of the Dyck language to include several different types of parentheses.

It seems to me that the first time the term “Dyck language” is used to describe this language (and its generalisation) is in (Chomsky, N.; Schützenberger, M. P. The algebraic theory of context-free languages. 1963 Computer programming and formal systems, pp. 118–161). Furthermore, all sources online agree that the “Dyck” in question is Walther von Dyck, who introduced the notion of a group presentation in 1882.

However, in the above paper, I can only see a weak reason as to why this language is named after von Dyck. A paragraph directly following the definition reads: The Dyck Language $$D_{2n}$$ on the $$2n$$ letters $$x_{pm i} : (1 leq i leq n)$$ (…) is a very familiar mathematical object: if $$varphi$$ is the homomorphism of the free monoid generated by $${ x_{pm i}}$$ onto the free group generated by the subset $${ x_i mid i > 0}$$ that satisfies identically $$(varphi x_i)^{-1} = varphi x_{-i}$$, then $$D_{2n}$$ is the kernel of $$varphi$$.

This alternate characterisation is obviously related to presentations, and thus has some connection with von Dyck. However, I am uncertain whether this is the full reason as to why it is named after him. Perhaps there is an intermediate study of the Dyck language inbetween the work of von Dyck and Chomsky-Schützenberger which makes this connection stronger? Thus, my question:

Why is the “Dyck language” named after von Dyck?

Of course, the same question might as well be asked about “Dyck paths” in combinatorics, closely related to the Catalan numbers, but it seems to me quite clear that Dyck paths were named after the Dyck language.

Any thoughts would be appreciated! Posted on Categories Articles