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  "body": "\n## Definition\n\nLet $\\ast$ be a [[binary operation|0M418FE7BB]] on a set $S$.\nThe *opposite* of $(S, \\ast)$ is $(S, \\ast^{\\mathrm{op}})$, where $a \\ast^{\\mathrm{op}} b = b \\ast a$ for all $a, b \\in S$.\nFor a structure with several operations, such as a ring, the opposite structure reverses the chosen multiplication and keeps all other operations and constants.\n\n## Remarks\n\nExample: if $(G, \\cdot)$ is a [[group|G5S1M9]], then $(G, \\cdot^{\\mathrm{op}})$ is a group with the same identity and the same inverses.\nAssociativity holds since $(a \\cdot^{\\mathrm{op}} b) \\cdot^{\\mathrm{op}} c = c \\cdot b \\cdot a = a \\cdot^{\\mathrm{op}} (b \\cdot^{\\mathrm{op}} c)$.\nThe map $x \\mapsto x^{-1}$ is an isomorphism from $G$ to its opposite, because $(xy)^{-1} = y^{-1} x^{-1}$.\n\nNon-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.\nThe transpose $A \\mapsto A^T$ is still an isomorphism from the matrix ring to its opposite, because $(AB)^T = B^T A^T$.\n\nCommon notations are $S^{\\mathrm{op}}$ and $S^{\\circ}$.\nThe opposite of the opposite is the original structure.\n",
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