# Definition:Induced Operation

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## Disambiguation

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**Induced Operation** may refer to:

- Operation Induced by Restriction: Where $\struct {T, \circ} \subseteq \struct {S, \circ}$, 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 $\struct {S, \circ}$ is an algebraic structure, the operation $\circ_\PP$ defined on the power set $\powerset S$ as:

- $A \circ_\mathcal P B = \set {a \circ b: a \in A, b \in B}$

- Operation Induced on Quotient Set: Where $\struct {S, \circ}$ is an algebraic structure and $\RR$ is a congruence relation on $\struct {S, \circ}$, the operation $\circ_\RR$ defined on the the quotient set $S / \RR$ as:

- $\eqclass x \RR \circ_\RR \eqclass y \RR = \eqclass {x \circ y} \RR$

- Operation Induced by Direct Product: Where $\displaystyle S = \prod_{k \mathop = 1}^n S_k$ is the cartesian product of the algebraic structures $\struct {S_1, \circ_1}, \struct {S_2, \circ_2}, \dotsc, \struct {S_n, \circ_n}$, the operation $\circ$ defined as:

- $\tuple {s_1, s_2, \dotsc, s_n} \circ \tuple {t_1, t_2, \dotsc, t_n} := \tuple {s_1 \circ_1 t_1, s_2 \circ_2 t_2, \dotsc, s_n \circ_n t_n}$ for all ordered $n$-tuples in $S$

- Pointwise Operation: Where $\struct {T, \circ}$ is an algebraic structure, $S$ is a set and $T^S$ is the set of all mappings from $S$ to $T$, the
**pointwise operation on $T^S$ induced by $\circ$**is defined on mappings $f$ and $g$ as:

- $f \oplus g: S \to T: \forall x \in S: \map {\paren {f \oplus g} } x = \map f x \circ \map g x$