Does Algebraic Structure (Set with $k$-ary Operation) Imply Each of its $k$ Domains and Codomain are Equal? An operation, in the most general sense, is a function which takes a cartesian product $S_1 \times S_2 \times \cdots \times S_k$ as its domain and another set $T$ as its codomain. In general, each of the $k$ sets can be different.
However, now let's talk about algebraic structures. Wikipedia (a bit carelessly) defines an Algebraic Structure as a set $S$ with one or more finitary operations "defined on it". What's unclear about this definition is what is meant by "defined on it".
Here is my question: does this definition of an algebraic structure $(S,*)$, where $*:S_1 \times S_2 \times \cdots \times S_k \to T$, implicitly imply that $S_1=S_2=\cdots=S_k$? Also, does it imply that $T=S$? If so, this should be explicitly said, because an "operation" in the general sense does not imply this. Maybe we can edit the Wikipedia page to be more clear.
Also, what word or term would one use to distinguish between an operation $*:S_1 \times S_2 \times \cdots \times S_k \to T$ which takes $S_1=S_2=\cdots=S_k=T$ and one which does not?
Thank you!  
 A: An algebraic structure is a tuple of the form $(S,f_i)_{i\in I}$ for some set $I$. Associated to each $i\in I$ there is a natural number $n_i$ (the arity of $f_i$) such that $f_i:S^{n_i}\to S$; that is, the domain of $f_i$ is a finite power of $S$ rather than a Cartesian product of perhaps different sets. Note that the codomain of $f_i$ is also required to be $S$. Here, $I$ could be empty (although the link you give requires that it is not), and some of the $n_i$ could be $0$ (in which case we could identify $f_i$ with the element of $S$ that $f_i$ "selects").
Although algebraic structures are ubiquitous in mathematics, universal algebra is the area that studies them (not quite abstract algebra, as suggested in the link you gave). 
Sometimes, particularly in set theory, one talks of "algebras" to mean structures $(S,f_i)_{i\in I}$ where, for all $i\in I$, $f_i:S^{<\mathbb N}\to S$. Here, $S^{<\mathbb N}$ is the set of finite tuples of elements of $S$. But note that each $f_i$ can be identified with countably many functions $f_{i,n}$, where $f_{i,n}:S^n\to S$ is simply $f_i\upharpoonright S^n$, so that this version is really just a particular case (in disguise) of the general notion. 
Finally, note that not all structures studied in mathematics are algebraic structures. For instance, metric spaces are pairs $(M,d)$ where $d:M\times M\to \mathbb R$, even if $\mathbb R$ and $M$ have nothing in common as sets.
