Opposite structure

Summary

The opposite of a structure with a binary operation $\ast$ has the same elements and the reversed operation $a \ast^{\mathrm{op}} b = b \ast a$. The opposite of a group is a group, and the opposite of a monoid is a monoid. The opposite ring reverses the multiplication and keeps the addition. Opposite structures turn left notions into right notions, for example left modules over a ring into right modules over its opposite ring. A structure equals its opposite exactly when its operation is commutative.