Definition:Induced Operation


 * Operation Induced by Restriction: Where $\left({T, \circ}\right) \subseteq \left({S, \circ}\right)$, the restriction of $\circ$ to $T$, namely $\circ \restriction_T$, is called the operation induced on $T$ by $\circ$.


 * Operation Induced on Power Set: Where $\left({S, \circ}\right)$ is an algebraic structure, the operation $\circ_\mathcal P$ defined on the power set $\mathcal P \left({S}\right)$ as:
 * $A \circ_\mathcal P B = \left\{{a \circ b: a \in A, b \in B}\right\}$


 * Operation Induced on Quotient Set: Where $\left({S, \circ}\right)$ is an algebraic structure and $\mathcal R$ is a congruence relation on $\left({S, \circ}\right)$, the operation $\circ_\mathcal R$ defined on the the quotient set $S / \mathcal R$ as:
 * $\left[\!\left[{x}\right]\!\right]_\mathcal R \circ_\mathcal R \left[\!\left[{y}\right]\!\right]_\mathcal R = \left[\!\left[{x \circ y}\right]\!\right]_\mathcal R$


 * Operation Induced by Direct Product: Where $\displaystyle S = \prod_{k=1}^n S_k$ is the cartesian product of the algebraic structures $\left({S_1, \circ_1}\right), \left({S_2, \circ_2}\right), \ldots, \left({S_n, \circ_n}\right)$, the operation $\circ$ defined as:
 * $\left({s_1, s_2, \ldots, s_n}\right) \circ \left({t_1, t_2, \ldots, t_n}\right) := \left({s_1 \circ_1 t_1, s_2 \circ_2 t_2, \ldots, s_n \circ_n t_n}\right)$ for all ordered $n$-tuples in $S$


 * Operation Induced on Set of Mappings: Where $\left({T, \circ}\right)$ is an algebraic structure, $S$ is a set and $T^S$ is the set of all mappings from $S$ to $T$, the operation on $T^S$ induced by $\circ$ is defined on mappings $f$ and $g$ as:
 * $f \oplus g: S \to T: \forall x \in S: \left({f \oplus g}\right) \left({x}\right) = f \left({x}\right) \circ g \left({x}\right)$