, ; 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. See also Uniqueness theorem Structural rigidity, a mathematical theory describing the degrees of freedom of ensembles of rigid physical objects connected together by flexible hinges. Level structure (algebraic geometry) References Mathematical terminology