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.