quasi-inverse of a function
Let be a function from sets to . A quasi-inverse of is a function such that
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where , and
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, where denotes functional composition operation.
Note that is the range of .
Examples.
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If is a real function given by . Then defined on and also defined on are both quasi-inverses of .
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If defined on . Then defined on is a quasi-inverse of . In fact, any where will do. Also, note that on is also a quasi-inverse of .
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If , the step function on the reals. Then by the previous example, , any , is a quasi-inverse of .
Remarks.
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Every function has a quasi-inverse. This is just another form of the Axiom of Choice. In fact, if , then for every subset of such that , there is a quasi-inverse of whose domain is .
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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 , then there are at least two quasi-inverses, one with domain and one with domain . So is onto. To see that is one-to-one, let be the quasi-inverse of . Now suppose . Let and assume . Define by if , and . Then is easily verified as a quasi-inverse of that is different from . This is a contradition. So . Similarly, and therefore .
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Conversely, if is a bijection, then the inverse of is a quasi-inverse of . In fact, has only one quasi-inverse.
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Let be a quasi-inverse of , then the restriction of to is one-to-one. If and are quasi-inverses of one another, and strictly includes , then is not one-to-one.
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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, as any quasi-inverse of a real function is also its pseudo-inverse as an element of the ring. Any space whose ring of continuous functions is Von Neumann regular is a P-space.
References
- 1 B. Schweizer, A. Sklar, Probabilistic Metric Spaces, Elsevier Science Publishing Company, (1983).
Title | quasi-inverse of a function |
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Canonical name | QuasiinverseOfAFunction |
Date of creation | 2013-03-22 16:22:14 |
Last modified on | 2013-03-22 16:22:14 |
Owner | CWoo (3771) |
Last modified by | CWoo (3771) |
Numerical id | 11 |
Author | CWoo (3771) |
Entry type | Definition |
Classification | msc 03E20 |
Synonym | quasi-inverse |
Defines | quasi-inverse function |