In mathematics, a rigid collection C of mathematical objects (for instance sets or functions) is one in which every c ∈ C is uniquely determined by less information about c than one would expect.
The above statement does not define a mathematical property; instead, it describes in what sense the adjective "rigid" is typically used in mathematics, by mathematicians.
Combinatorial use
In combinatorics, the term rigid is also used to define the notion of a rigid surjection, which is a surjection for which the following equivalent conditions hold:[1]
- For every , ;
- Considering as an -tuple , the first occurrences of the elements in are in increasing order;
- maps initial segments of to initial segments of .
This relates to the above definition of rigid, in that each rigid surjection uniquely defines, and is uniquely defined by, a partition of into pieces. Given a rigid surjection , the partition is defined by . Conversely, given a partition of , order the by letting . If is now the -ordered partition, the function defined by is a rigid surjection.