quasi-inverse of a function
where , and
, where denotes functional composition operation.
Note that is the range of .
If is a real function given by . Then defined on and also defined on are both quasi-inverses of .
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 .
If , the step function on the reals. Then by the previous example, , any , is a quasi-inverse of .
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 .
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 .
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.
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.
- 1 B. Schweizer, A. Sklar, Probabilistic Metric Spaces, Elsevier Science Publishing Company, (1983).
|Title||quasi-inverse of a function|
|Date of creation||2013-03-22 16:22:14|
|Last modified on||2013-03-22 16:22:14|
|Last modified by||CWoo (3771)|