I want to define the set $F$ of functions on a set $X$ that can be generated from a finite number of selected operations defined on $X$ (the set from which these operations are selected need not be finite, but each function needs to expressible using a finite number of operations).
At first I had considered a subset of the infinite product...
$$F(X)\subseteq X^X=\prod_{x\in X}X$$
...for which $f\in F(X)$ is a function (whose value at $x$ is $f(x)=f_x$) but then I realized that said product does not include obvious cases where a function is undefined at a point in its domain, for example $f(0)$ where $f:\mathbb{C}\to\mathbb{C};\ f(z)=1/z$.
The easiest solution I could think of was to define a 'nonelement' $\bot$ so that if $f:X\to X$, then $f(x)\ \text{undefined}\iff f(x)=\bot$. This is basically saying that the set of solutions $y$ to the equation $y=f(x)$ is empty. For example, I might have $f:\mathbb{C}\to\mathbb{C};\ f(z)=1/z\implies f(0)=\bot$.
Then I can modify the product accordingly to include functions which are undefined at some point...
$$F(X)\subseteq\prod_{x\in X}X\cup\{\bot\}$$
Obviously, there are some problems with having such a 'null element' a few of which were addressed in this question, but none of them are too daunting on their own. The real difficulty is getting various definitions to work together. I can see a lot of use for a null element, so imagine someone has already done this. Is there an accepted convention for such an element? are there any sources which look at it in detail?
Abstract/universal algebra tag is for context - If $(X,S_{op})$ is the algebraic structure consisting of the set $X$ and the collection of operations $S_{op}$, then $F(X)$ as described above is the equivalent of the class of elementary functions in $(X,S_{op})$.