---
tag: D33V8HE0KY
kind: concept
name: Opposite structure
msc: [08A02, 20A05]
sources:
  - url: https://en.wikipedia.org/wiki/Opposite_group
---

## Summary

The opposite of a structure with a [[binary operation|0M418FE7BB]] $\ast$ has the same elements and the reversed operation $a \ast^{\mathrm{op}} b = b \ast a$.
The opposite of a [[group|G5S1M9]] is a group, and the opposite of a [[monoid|0N6R902FG1]] is a monoid.
The [[opposite ring|M8Q24XB0AF]] 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|099E72KTF8]].

## Explanations

### L1 human

The opposite structure reads every product backwards.
If $a$ then $b$ gives $a \ast b$, in the opposite structure $a$ then $b$ gives $b \ast a$.
For the composition of moves of a puzzle, the opposite means doing the moves in the other order.
For addition of numbers, nothing changes, because $a + b = b + a$.