Opposite structure

Remarks

Example: if $(G, \cdot)$ is a group, then $(G, \cdot^{\mathrm{op}})$ is a group with the same identity and the same inverses. Associativity holds since $(a \cdot^{\mathrm{op}} b) \cdot^{\mathrm{op}} c = c \cdot b \cdot a = a \cdot^{\mathrm{op}} (b \cdot^{\mathrm{op}} c)$. The map $x \mapsto x^{-1}$ is an isomorphism from $G$ to its opposite, because $(xy)^{-1} = y^{-1} x^{-1}$.

Non-example: for $2 \times 2$ real matrices, the opposite multiplication is not the same operation as the usual one, because the matrices do not all commute. The transpose $A \mapsto A^T$ is still an isomorphism from the matrix ring to its opposite, because $(AB)^T = B^T A^T$.

Common notations are $S^{\mathrm{op}}$ and $S^{\circ}$. The opposite of the opposite is the original structure.