Set-theoretic solutions were studied by Drinfeld. In this case, there is an R -matrix invariant basis X for the vector space V in the sense that the R -matrix maps the induced basis on V ⊗ ⊗ V to itself. This then induces a map r : X × × X → → X × × X given by restriction of the R -matrix to the basis. The set-theoretic Yang–Baxter equation is then defined using the 'twisted' alternate form above, asserting ( i d × × r ) ( r × × i d ) ( i d × × r ) = ( r × × i d ) ( i d × × r ) ( r × × i d ) as maps on X × × X × × X . The equation can then be considered purely as an equation in the category of sets.