Definition 6

A subset $A$ of a set $E$ is called stable under the law of composition $T$ on $E$ if the composition of two elements of $A$ belongs to $A$. The mapping $(x, y) \mapsto x T y$ of $A \times A$ into $A$ is then called the law induced on $A$ by the law $T$. The set $A$ with the law induced by $T$ is called a submagma of $E$.

In other words, for $A$ to be stable under a law $T$ it is necessary and sufficient that $A T A \subset A$. A stable subset of $E$ and the corresponding submagma are often identified.

The intersection of a family of stable subsets of $E$ is stable; in particular there exists a smallest stable subset $A$ of $E$ containing a given subset $X$; it is said to be generated by $X$ and $X$ is called a generating system of $A$ or a generating set of $A$. The corresponding submagma is also said to be generated by $X$.