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

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.

• To be honest I've never seen the notation $\prod x$ without indices. Can you provide the definition of $\prod x$, for an arbitrary $x$? – Git Gud Oct 6 '14 at 18:27
• Or the definition of $\prod\langle \mathbf{A}_i \ | \ i \in I \rangle$. – Git Gud Oct 6 '14 at 18:37
• 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. – Andrés E. Caicedo Oct 6 '14 at 18:46
• @Git Gut. $\prod \langle \mathbf{A}_i \ | \ i \in I \rangle$ denotes the same object as $\prod_{i \in I} \mathbf{A}_i$. – Roy Oct 6 '14 at 18:52
• 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). – Git Gud Oct 6 '14 at 19:26

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.