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[parent] quasi-inverse of a function (Definition)

Let $ f:X\to Y$ be a function from sets $ X$ to $ Y$. A quasi-inverse $ g$ of $ f$ is a function $ g$ such that

  1. $ g:Z\to X$ where $ \operatorname{ran}(f)\subseteq Z\subseteq Y$, and
  2. $ f\circ g\circ f=f$, where $ \circ$ denotes functional composition operation.

Note that $ \operatorname{ran}(f)$ is the range of $ f$.

Examples.

  1. If $ f$ is a real function given by $ f(x)=x^2$. Then $ g(x)=\sqrt{x}$ defined on $ [0,\infty)$ and $ h(x)=-\sqrt{x}$ also defined on $ [0,\infty)$ are both quasi-inverses of $ f$.
  2. If $ f(x)=1$ defined on $ [0,1)$. Then $ g(x)=\frac{1}{2}$ defined on $ \mathbb{R}$ is a quasi-inverse of $ f$. In fact, any $ g(x)=a$ where $ a\in [0,1)$ will do. Also, note that $ h(x)=x$ on $ [0,1)$ is also a quasi-inverse of $ f$.
  3. If $ f(x)=[x]$, the step function on the reals. Then by the previous example, $ g(x)=[x]+a$, any $ a\in[0,1)$, is a quasi-inverse of $ f$.

Remarks.

  • Every function has a quasi-inverse. This is just another form of the Axiom of Choice. In fact, if $ f:X\to Y$, then for every subset $ Z$ of $ Y$ such that $ \operatorname{ran}(f)\subseteq Z$, there is a quasi-inverse $ g$ of $ f$ whose domain is $ Z$.
  • However, a quasi-inverse of a function is in general not unique, as illustrated by the above examples. When it is unique, the function must be a bijection:
    If $ \operatorname{ran}(f)\ne Y$, then there are at least two quasi-inverses, one with domain $ \operatorname{ran}(f)$ and one with domain $ Y$. So $ f$ is onto. To see that $ f$ is one-to-one, let $ g$ be the quasi-inverse of $ f$. Now suppose $ f(x_1)=f(x_2)=z$. Let $ g(z)=x_3$ and assume $ x_3\ne x_1$. Define $ h:Y\to X$ by $ h(y)=g(y)$ if $ y\ne z$, and $ h(z)=x_1$. Then $ h$ is easily verified as a quasi-inverse of $ f$ that is different from $ g$. This is a contradition. So $ x_3=x_1$. Similarly, $ x_3=x_2$ and therefore $ x_1=x_2$.
  • Conversely, if $ f$ is a bijection, then the inverse of $ f$ is a quasi-inverse of $ f$. In fact, $ f$ has only one quasi-inverse.
  • The relation of being quasi-inverse is not symmetric. In other words, if $ g$ is a quasi-inverse of $ f$, $ f$ need not be a quasi-inverse of $ g$. In the second example above, $ h$ is a quasi-inverse of $ f$, but not vice versa: $ h(0)=0$, but $ hfh(0)=hf(0)=h(1)=1\ne h(0)$.
  • Let $ g$ be a quasi-inverse of $ f$, then the restriction of $ g$ to $ \operatorname{ran}(f)$ is one-to-one. If $ g$ and $ f$ are quasi-inverses of one another, and $ \operatorname{g}$ strictly includes $ \operatorname{ran}(f)$, then $ g$ is not one-to-one.
  • The set of real functions, with addition defined element-wise and multiplication defined as functional composition, is a ring. By remark 2, it is in fact a Von Neumann regular ring. Any space whose ring of continuous functions is Von Neumann regular is a P-space.

Bibliography

1
B. Schweizer, A. Sklar, Probabilistic Metric Spaces, Elsevier Science Publishing Company, (1983).



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Other names:  quasi-inverse
Also defines:  quasi-inverse function

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Cross-references: von Neumann regular, ring of continuous functions, von Neumann regular ring, ring, strictly, restriction, symmetric, relation, inverse, one-to-one, onto, bijection, domain, subset, axiom of choice, reals, step function, real function, range, operation, composition, functional, function
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This is version 6 of quasi-inverse of a function, born on 2006-11-03, modified 2007-05-16.
Object id is 8509, canonical name is QuasiInverseOfAFunction.
Accessed 1953 times total.

Classification:
AMS MSC: 03E20 (Mathematical logic and foundations :: Set theory :: Other classical set theory )
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