While reading about algebras and coalgebras, I keep coming across two definitions of an algebra $A$. One definition uses the Cartesian product $\times$, while another uses the tensor product $\otimes$. (The definition for a coalgebra always uses $\otimes$.) Both definitions are ones involving commutativity of diagrams.
When should $\times$ be used, and when should $\otimes$ be used? Is the reason for $\otimes$ due to the $K$-linearity of the underlying maps (where $K$ is a field), while we do not require $K$-linearity for the definition involving $\times$? I suppose I'm trying to make sense of which definition should use $\times$ (and why) and which should use $\otimes$. Could someone please explain these two variations to me?
The universal property of the tensor product of vector spaces says $$\text{Hom}_{K}(A\otimes B,C)\cong\text{Bil}_{K}(A,B;C)$$ so in the case that $A=B=C$, this means that bilinear functions $$C\times C \to C$$ are in one-to-one correspondence with linear functions $$C\otimes C \to C$$ so in the definition of an algebra you can ask for any of those, but as I said they are completely equivalent.
In the case of a coalgebra, you can only talk about a linear function $$C\to C\otimes C$$ since there is no equivalent way to say this in terms of $\times$.