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What is the product of an empty family of similiar algebras, that is $\prod\langle \mathbf{A}_i \mid i \in I \rangle $, where $I = \emptyset$?

The family $\langle \mathbf{A}_i \mid i \in \emptyset \rangle $ equals $\emptyset$ beacuse the function $i \colon \emptyset \mapsto X$ (whatever the $X$ is) is the empty function $\emptyset$.

So the question is: what is $\prod \emptyset$? Is it $\{ \emptyset \}$?

Am I right? I believe it is, but I haven't a step-by-step explanation. What is the universum of such product? How does a function interpretation look like (and more important: why?)

Thanks.

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  • $\begingroup$ To be honest I've never seen the notation $\prod x$ without indices. Can you provide the definition of $\prod x$, for an arbitrary $x$? $\endgroup$ – Git Gud Oct 6 '14 at 18:27
  • $\begingroup$ Or the definition of $\prod\langle \mathbf{A}_i \ | \ i \in I \rangle$. $\endgroup$ – Git Gud Oct 6 '14 at 18:37
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    $\begingroup$ Yes. As a set, the product is the collection of all functions $f$ with domain $I$ such that for all $i\in I$ we have $f(i)\in|\mathbf A_i|$, where $|\mathbf M|$ is the universe of the algebra $\mathbf M$. If $I=\emptyset$, the only such function is the empty function. $\endgroup$ – Andrés E. Caicedo Oct 6 '14 at 18:46
  • $\begingroup$ @Git Gut. $\prod \langle \mathbf{A}_i \ | \ i \in I \rangle$ denotes the same object as $\prod_{i \in I} \mathbf{A}_i$. $\endgroup$ – Roy Oct 6 '14 at 18:52
  • $\begingroup$ Then it's as Andres said. See the wikipedia entry on this. Which isn't surprising considering that the empty sum yields its neutral element, so does the empty product (since $\{\varnothing \}$=1). $\endgroup$ – Git Gud Oct 6 '14 at 19:26
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To cut down on the number of unanswered questions, here we go.

Given any set of sets $\{B_i\}_{i\in I},$ we have by definition that $$\prod_{i\in I}B_i:=\left\{f:I\to\bigcup_{i\in I}B_i\mid\forall i\in I,f(i)\in B_i\right\},$$ so when $I=\emptyset,$ this is clearly just the set containing the empty function $\emptyset,$ and nothing else.

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