definition of composition of maps
Topics
About: set
The table of contents of this article
Starting Context
- The reader knows a definition of map.
Target Context
- The reader will have a definition of composition of maps.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( \{S'_1, ..., S'_n\}\): \(\subseteq \{\text{ the sets }\}\)
\( \{S_2, ..., S_{n + 1}\}\): \(\subseteq \{\text{ the sets }\}\), such that \(\forall j \in \{2, ..., n\} (S_j \subseteq S'_j)\)
\( \{f_1, ..., f_n\}\): \(f_j: S'_j \to S_{j + 1}\), \(\in \{\text{ the maps }\}\)
\(*f_n \circ ... \circ f_1\): \(S'_1 \to S_{n + 1}, s \mapsto f_n (f_{n - 1} (... (f_1 (s))))\)
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Conditions:
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2: Note
A point is that for each \(j \in \{2, ..., n\}\), \(S_j \subseteq S'_j\) is required.
That is because if \(S_j \subseteq S'_j\) did not hold, \(f_j (f_{j - 1} (... (f_1 (s))))\) would not be defined in general because \(f_{j - 1} (... (f_1 (s))) \in S'_j\) would not be guaranteed in general.
You may say that it should be OK if \(Ran (f_{j - 1}) \subseteq S'_j\), but in that case, you should be able to define another \(S_j\) such that \(Ran (f_{j - 1}) \subseteq S_j \subseteq S'_j\) instead. So, there is no restriction by this definition.
Another point is that \(S_j = S'_j\) is not required, so, you need to take care not to fall into some pitfalls: for example, the proposition that a finite composition of surjections is not necessarily any surjection.
Another point is that \(S_j \subseteq S'_j\) is just as sets: for example, when \(S'_j\) is a vectors space, \(S_j\) may not be any vectors subspace of \(S'_j\): refer to the proposition that for any linear map between any modules and any linear map from any supermodule of the codomain of the 1st map into any module, the composition of the 2nd map after the 1st map is linear.
As an important property, composition of maps is associative: \(f_3 \circ (f_2 \circ f_1) = (f_3 \circ f_2) \circ f_1\), which guarantees any association by the proposition that for any structure, the associativity for any 3 items allows any association.
That is because \((f_3 \circ (f_2 \circ f_1)) (s) = f_3 ((f_2 \circ f_1) (s)) = f_3 (f_2 (f_1 (s)))\) while \(((f_3 \circ f_2) \circ f_1) (s) = (f_3 \circ f_2) (f_1 (s)) = f_3 (f_2 (f_1 (s)))\).
