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.

Some examples include:

In combinatorics, the term rigid is also used to define the notion of a rigid surjection, which is a surjection ${\displaystyle f:n\to m}$ for which the following equivalent conditions hold:

This relates to the above definition of rigid, in that each rigid surjection ${\displaystyle f}$ uniquely defines, and is uniquely defined by, a partition of ${\displaystyle n}$ into ${\displaystyle m}$ pieces. Given a rigid surjection ${\displaystyle f}$, the partition is defined by ${\displaystyle n=f^{-1}(0)\sqcup \cdots \sqcup f^{-1}(m-1)}$. Conversely, given a partition of ${\displaystyle n=A_{0}\sqcup \cdots \sqcup A_{m-1}}$, order the ${\displaystyle A_{i}}$ by letting ${\displaystyle A_{i}\prec A_{j}\iff \min A_{i}<\min A_{j}}$. If ${\displaystyle n=B_{0}\sqcup \cdots \sqcup B_{m-1}}$ is now the ${\displaystyle \prec }$-ordered partition, the function ${\displaystyle f:n\to m}$ defined by ${\displaystyle f(i)=j\iff i\in B_{j}}$ is a rigid surjection.

This article incorporates material from rigid on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.