First of all apologies because I'm not a mathematician. So let's suppose I have a number of sets $A_1,...,A_n$ and functions $f_i\colon A_i \to A_{i+1}$ with $1\le i<n$. I want to know how to notate a mapping exists from $A_1 \to A_n$ through the composition of partial functions $f_1\circ...\circ f_{n-1}$.
Note that in the above image there is a mapping from $a_{11} \to a_{42}$, $a_{12} \to a_{42}$, $a_{14} \to a_{44}$ but not from $a_{13}$ to any element in set $A_4$.
The context in which I would like to check such a mapping is so I can use it as a condition perhaps like so:
$f(a) = (f_1 \circ...\circ f_{n-1})(a) \\ f(a) = f \colon A^\prime_1 \to A_n \\ A^\prime_1 \subseteq A_1$
Note $A^\prime_1$ is used because $f(a)$ is a partial function.
EDIT before I used $\exists (f_{n-1}\circ ...\circ f_1)(a): A_1\rightarrow A_n$ in the if part of the conditional so I changed it to another example I could think of:
$b = \begin{cases}1 &\text{if } a \in A^\prime_1 \\ 0 &\text{otherwise} \end{cases}$
Questions:
- Is there some standard notation for checking such a mapping?
- If not, has what I written made sense or are there some more familiar or appropriate methods I could use?
The symbol "$\downarrow$" is often used to indicate defined-ness of a partial function on a given input. So e.g. you could abbreviate the condition by "$f_{n-1}\circ...\circ f_1(a)\downarrow$." I don't know of any shorter expression for it.
(Incidentally, if this is a condition you're interested in it may be worth doing a bit more work and letting (say) $g_{i,j}$ for $i\le j$ denote the function $f_{j-1}\circ...\circ f_i$; then your condition is just $g_{1, n}(a)\downarrow$. Introducing these new functions may or may not be worthwhile depending on what you're interested in.)