with the understanding that is undefined for and is undefined if . We define an equivalence relation on as follows: for elements we have if and only if there exists a finite sequence of applications of the and functions starting with and ending with , where the empty sequence is allowed. That the empty sequence is allowed accounts for the reflexivity of . Symmetryfollows from the observation that any finite sequence of applications of to a string can be undone with a finite sequence of applications of . Transitivity is clear from the definition.
The equivalence relation partitions the language into equivalence classes. If we take to denote the empty string, then the language corresponding to the equivalence class is called the Dyck language.
We can define an equivalence relation on the Dyck language . For we have if and only if , i.e. and have the same length. This relation partitions the Dyck language where . Note that is empty for odd . E.g. there are 14 words in , i.e. [[[]]], [[]], [[]], [[]], [[]], [], [[]], [], [][], [], [], [], [], .
Having introduced the Dyck words of length , we can introduce a relationship on them. For every we define a relation on ; for we have if and only if can be reached from by a series of proper swaps. A proper swap in a word swaps an occurrence of '][' with ''. For each the relation makes into a partially ordered set. The relation is reflexive because an empty sequence of proper swaps takes to . Transitivity follows because we can extend a sequence of proper swaps that takes to by concatenating it with a sequence of proper swaps that takes to forming a sequence that takes into . To see that is also antisymmetric we introduce an auxiliary function where ranges over all prefixes of . The following table illustrates that is strictly monotonic with respect to proper swaps.
Strict monotonicity of
partial sums of
partial sums of
Difference of partial sums
Hence so when there is a proper swap that takes into . Now if we assume that both and , then there are non-empty sequences of proper swaps such is taken into and vice versa. But then which is nonsensical. Therefore, whenever both and are in , we have , hence is antisymmetric.
The partial ordered set is shown in the illustration accompanying the introduction if we interpret a [ as going up and ] as going down.