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A binary operation \(*\) on a set \(S\) is a function mapping \(S \times S\) into \(S\).
e.g.
一般的 \(+\) (addition) 是 \(\mathbb{R}\) 上的 binary operation；但 \(/\) (division) 不是，因為 \(a/0\) 未定義。
Let \(*\) be a binary operation on a set \(S\) and let \(H\) be a subset of \(S\). If for all \(a, b \in H\) we also have \(a ∗ b \in H\), then \(H\) is closed under \(*\).
In this case, the binary operation on \(H\) given by restricting \(∗\) to \(H\) is the induced operation of \(*\) on \(H\).
e.g.
We have \(\mathbb{R} \subset \mathbb{C}\). The addition \(+\) on \(\mathbb{C}\) induces a binary operation on \(\mathbb{R}\).