Examples of partial functions outside recursive function theory?

Question

My math background is very narrow. I've mostly read logic, recursive function theory, and set theory.

In recursive function theory one studies partial functions on the set of natural numbers.

Are there other areas of mathematics in which (non-total) partial functions are important? If so, would someone please supply some references?

Thanks!

Answer 1

In functional analysis, the concept of an unbounded operator is closely connected with partial functions. The natural examples of unbounded operators are linear operators that are defined only on a dense proper subspace of a Banach space. For example, the "derivative" operator is an unbounded linear operator on the space $L_2[0,1]$, but it is far from being total. The use of partial functions turns out to be vital; for example the Hellinger–Toeplitz theorem is often interpreted as saying that it is necessary to consider partial operators in order to formalize quantum mechanics.

Answer 2

Sure. For example, meromorphic functions in complex analysis are partial functions. Multiplicative inverse is another partial function (say on a ring).

It is worth mentioning that mathematical tradition in most areas (in my experience) is to refer to a partial function as a total function by restricting the domain and not to actually talk about partial functions. For example, most people talk about multiplicative inverse as a total function, but defined on the group of units of a ring.

This works fine but I think it is unnecessary. There are many places in mathematics where it would be more natural to use partial functions where people don't. For example, composition of morphisms in a category is a partially defined operation on pairs of morphisms, but that's not how it gets formalized: instead it is formalized as a family of totally defined operations $\text{Hom}(a, b) \times \text{Hom}(b, c) \to \text{Hom}(a, c)$.

Answer 3

In one sense, surely, it is deeply important that the square root function with domain and co-domain the positive rationals $\mathbb{Q}$ is partial (as are the cube root function, fourth root function, etc. etc.). That's the non-trivial ancient discovery that leads us to introduce the concept of irrational numbers.

Likewise, it is important the square root function etc. with domain and co-domain the reals $\mathbb{R}$ is partial. Wanting an algebraically complete field we introduce the concept of complex numbers.

Of course those cases are very familiar, but that surely does not make them unimportant! (And note that these cases are not met by restricting the domain to avoid partiality, but -- in the first place -- by augmenting the codomain.)

Answer 4

I think that the reason you have this particular interests in partial function or even heard of the term is mostly because you have read recursion theory.

To elaborate, in most other areas of mathematics, function are by definition totally defined on their domain. To say that $f : X \rightarrow Y$ but $f$ is not define on all of $X$ is not a usual mathematical practice. Most people would restrict the domain so that $f$ is totally defined. For example, the function $f(x) = \frac{1}{x}$ where $x$ comes from some field. $f : \mathbb{R} \rightarrow \mathbb{R}$ then $f$ would be partial. Most people would likely modify this to say $f : \mathbb{R} - \{0\} \rightarrow \mathbb{R}$. In other areas, people would likely be careful to specify the domain and range.

Partial functions do appear in other areas of mathematics. Qiaochu Yuan seem to mention the study of the poles of complex value functions. For example, the residue formula gives some useful information about complex value functions using the poles.

However, in computability theory or recursion theory, there is a useful significance to introduce total and partial functions. In computability theory, you study subsets of the natural numbers and function defined on these subset. Because these function and sets correspond to Turing Machine and algorithms, their inputs (for simplicity) can be coded as natural numbers. By specifying the domain of function, in the context of recursion theory, to be $\omega$, represents the intuitive idea of an algorithm. Moreover, many computer programs, Turing machines, or algorithm used in real practice don't terminate on all inputs. Because recursion theory is the study of the computational aspect of sets, it makes sense to include these sort of partial functions.

The above is sort of an intuitive idea of why recursion theory (the study of computation) should naturally consider partial functions because algorithms naturally don't halt on all input.

Moreover, partial functions play a major role in recursion theory that is not seen in many other areas of mathematics. The partial functions are essential to computability theory. First, you are aware of the enumeration theorem for partial computable functions (i.e. the universal Turing Machine exists). Almost every theorem in computability theory uses this fact. Moreover, it can be prove that there there is no effective enumeration of the computable functions. This enumeration is absolutely necessary in many of the construction such as finite injury arguments. Moreover, the c.e. sets play a very important role in computability theory. They are defined to be the domain of the partial computable functions. Many natural problems in mathematics such that Halting Problem, diophantine equations, etc have a correspond sets that is c.e.

The term partial function is more ubiquitous in computability theory because the most fundamental objects of the theory the enumeration theorem and the c.e. sets are naturally expressed using them.