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- Jun 22, 2012

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My issue is understanding the notation on Section 1-2, subsection 1-2.1 (see attachment).

Dauns is dealing with the product [TEX] \Pi \{ M_i | i \in I \} \equiv \Pi M_i [/TEX] and states in (ii) - see attachment

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Alternatively, the product can be viewed as consisting of all strings or sets

[TEX] x = \{ x_i | i \in I \} \equiv (x_i)_{i \in I} \equiv (x_i) \equiv ( \_ \_ \_ \ , x_i , \_ \_ \_ ) , x_i \in M_i; [/TEX] i-th

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I am not sure of the meaning of the above set of equivalences. Can someone briefly elaborate ... preferably with a simple example

If we take the case of I = {1,2,3} and consider the product [TEX] M_1 \times M_2 \times M_3 [/TEX] then does Dauns notation mean

[TEX] x = (x_1, x_2, x_3) [/TEX] where order in the triple matters (mind you if it does what are we to make of the statement [TEX] x = \{ x_i | i \in I \} [/TEX]

Can someone confirm that [TEX] x = (x_1, x_2, x_3) [/TEX] is a correct interpretation of Dauns notation?

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Dauns then goes on to define the direct sum as follows:

The direct sum [TEX] \oplus \{ M_i | i \in I \} \equiv \oplus M_i [/TEX] is defined as the submodule [TEX] \oplus M_i \subseteq \Pi M_i [/TEX] consisting of those elements [TEX] x = (x_i) \in \Pi M_i [/TEX] having at most a finite number of non-zero coordinates or components. Sometimes [TEX] \oplus M_i , \Pi M_i [/TEX] are called the external direct sum and the external direct product respectively.

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Can someone point out the difference between [TEX] \oplus M_i , \Pi M_i [/TEX] in the case of the example involving [TEX] M_1, M_2, M_3 [/TEX] - I cannot really see the difference! For example, what elements exactly are in [TEX] \Pi M_i [/TEX] that are not in [TEX] \oplus M_i [/TEX]

I would be grateful if someone can clarify these issues.

Peter

This has also been posted on MHF