Definition:Left Inverse Mapping
This page is about Left Inverse Mapping in the context of Mapping Theory. For other uses, see Left Inverse.
Definition
Let $S, T$ be sets where $S \ne \O$, that is, $S$ is not empty.
Let $f: S \to T$ be a mapping.
Let $g: T \to S$ be a mapping such that:
- $g \circ f = I_S$
where:
- $g \circ f$ denotes the composite mapping $f$ followed by $g$;
- $I_S$ is the identity mapping on $S$.
Then $g: T \to S$ is called a left inverse (mapping).
Examples
Inclusion of Reals in Complex Plane
Let $i_\R: \R \to \C$ be the inclusion mapping of the real numbers into the complex plane:
- $\forall x \in \R: \map {i_\R} z = x + 0 i$
From Inclusion Mapping is Injection, $i_\R$ is an injection.
Hence it has a left inverse $g: \C \to \R$ which, for example, can be defined as:
- $\forall z \in \C: \map g z = \map \Re z$
This left inverse is not unique.
For example, the mapping $h: \R \to \C$ defined as:
- $\forall z \in \C: \map g z = \map \Re z + \map \Im z$
is also a left inverse.
Also see
- Injection iff Left Inverse, which demonstrates that $g$ can not be defined unless $f$ is an injection.
- Results about left inverse mappings can be found here.
In the context of abstract algebra:
from which it can be seen that a left inverse mapping can be considered as a left inverse element of an algebraic structure whose operation is composition of mappings.
Sources
- 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 1.3$: Exercise $5$
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{I}$: Sets and Functions: Composition of Functions
- 1968: Ian D. Macdonald: The Theory of Groups ... (previous) ... (next): Appendix: Elementary set and number theory
- 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 2$: Sets and functions: Inverse images and inverse functions
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.3$: Mappings: Exercise $6$
- 1996: H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability ... (previous) ... (next): Appendix $\text{A}.7$: Inverses
- 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 2$: Functions: Exercise $2.5$