# Definition:Induced Operation

Jump to navigation
Jump to search

## Disambiguation

This page lists articles associated with the same title. If an internal link led you here, you may wish to change the link to point directly to the intended article.

**Induced Operation** may refer to:

- 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:

- $f \oplus g: S \to T: \forall x \in S: \left({f \oplus g}\right) \left({x}\right) = f \left({x}\right) \circ g \left({x}\right)$