Definition. Let $f:A\to B$ and $g:C\to D$ be partial functions. $g$ is said to be a subfunction of $f$ if $$g\subseteq f \cap (C\times D).$$

In other words, $g$ is a subfunction of $f$ iff whenever $x\in C$ such that $g(x)$ is defined, then $x\in A$ , $f(x)$ is defined, and $g(x)=f(x)$ .

If we set $C'=A\cap C$ and $D'=B\cap D$ , then $g\subseteq f\cap (C'\times D')$ , so there is no harm in assuming that $C$ and $D$ are subsets of $A$ and $B$ respectively, which we will do for the rest of the discussion.

In practice, whenever $g$ is a subfunction of $f$ , we often assume that $g$ and $f$ have the same domain and codomain. Otherwise, we would specify that $g$ is a subfunction of $f:A\to B$ with domain $C$ and codomain $D$ .

For example, $f:\mathbb{R} \to \mathbb{R}$ defined by $$f(x)=\sqrt{x^2-1}$$ is a partial function, whose domain of definition is $(-\infty,-1]\cup [1,\infty)$ , and the partial function $g:\mathbb{R} \to \mathbb{R}$ given by $$g(x)=\displaystyle{\frac{x^2-1}{\sqrt{x^2-1}}}$$ is a subfunction of $f$ . The domain of definition of $g$ is $(-\infty,-1)\cup (1,\infty)$ .

Two immediate properties of a subfunction $g:C\to D$ of $f:A\to B$ are

• the range of $g$ is a subset of the range of $f$ : $$g(C)\subseteq f(C),$$
• the domain of definition of $g$ is a subset of the domain of definition of $f$ : $$g^{-1}(D)\subseteq f^{-1}(D).$$

Definition. A subfunction $g:C\to D$ of $f:A\to B$ is called a restriction of $f$ relative to $D$ , if $g(C)=f(C)\cap D$ , and a restriction of $f$ if $g(C)=f(C)$ .

Every partial function $g:C\to D$ corresponds to a unique restriction $g':C\to g(C)$ of $g$ .

A restriction $g:C\to D$ of $f:A\to B$ is certainly a restriction of $f$ relative to $D$ , since $f(C)\cap D = g(C)\cap D = g(C)$ , but not conversely. For example, let $A$ be the set of all non-negative integers and $-_A: A^2\to A$ the ordinary subtraction. $-_A$ is easily seen to be a partial function. Let $B$ be the set of all positive integers. Then $-_B:B^2\to B$ is a restriction of $-_A:A^2\to A$ , relative to $B$ . However, $-_B$ is not a restriction of $-_A$ , for $n -_B n$ is not defined, while $n -_A n = 0\in A$ .

Bibliography

1

G. Grätzer: Universal Algebra, 2nd Edition, Springer, New York (1978).