definition of direct product of structures
Topics
About: structure
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
- 3: Note
Starting Context
- The reader knows a definition of structure.
- The reader knows a definition of product set.
Target Context
- The reader will have a definition of direct product of structures.
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:
\( \{S_\alpha\}\): \(S_\alpha \in \{\text{ the structures }\}\) with any common collection of operations, where \(\alpha \in A\) where \(A\) is any possibly uncountable index set
\(*\times_{\alpha} S_\alpha\): \(= \text{ the product set }\) with the collection of operations
//
Conditions:
For each operation, operation on products is product of operations
//
2: Natural Language Description
For any possibly uncountable number of structures with any common collection of operations, \(\{S_\alpha \vert \alpha \in A\}\), where \(A\) is a possibly uncountable index set, the product set, \(*\times_{\alpha \in A} S_\alpha\), with the collection of operations such that for any operation, operation on products is product of operations
3: Note
Although Description may be confusing as it tries to be general, the structures are for example some groups, which have the common multiplication operation (of course, each operation of each group is different from the operations of the other groups, but the operations are of the same kind, which we mean), and \(\times_{\alpha} p_\alpha \times_{\alpha} p'_\alpha = \times_{\alpha} p_\alpha p'_\alpha\), and the product is a group.
When \(S_\alpha\) is \(\mathbb{R}\) as the field, the collection of operations consists of \(+, *\), and when \(A\) is finite, it is like \((p_1, ..., p_n) * (p'_1, ..., p'_n) + (p''_1, ..., p''_n) = (p_1 * p'_1 + p''_1, ..., p_n * p'_n + p''_n)\).
Compare with 'direct sum of modules', which is defined only for modules and enforces the condition that only finite of \(\{p_\alpha \vert \alpha \in A\}\) are not \(0\).
