We have the definition of Cartesian Product as follows:
Let $X$ and $Y$ be two sets, then the set $X \times Y = \{(a, b) : a \in X,~b \in Y\}$, is called Cartesian product of $X$ and $Y$
Elements of $X \times Y$ are called ordered pairs. Any subset of $X \times Y$ is called a relation. If we take the special case when $Y = X$, we then have the ordered pairs of $X \times X.$ There are different terminologies are defined for different relations (i.e. subsets) of $X \times X.$ Out of them the equivalence relation which is the combination of three different relations (reflexive, symmetric, transitive) is one of the most influential one. Now, take a relation $S$ on $X$ (i.e. $S \subseteq X \times X$) that forms an equivalence relation and fix an element $a\in X$. Collect all elements (i.e. ordered pairs) containing $a$. Then the set defined and denoted as $cl(a) = \{x \in X \mid (x, a) \in S\}$ is called equivalence class of $a \in A$. This eventually leads to partition of the set. The partition can be defined as follows:
Let $A$ be a set and $A_1, A_2, \ldots, A_n$ are subsets of $A$ then the collection of these subsets defines a partition of $A$ if (i) they are disjoint i.e. $A_i \cap A_j = \phi,~\forall i \neq j$ (ii) $A = \cup_{i}^{n} A_i$
The main reason that I have mentioned all these is to express the influence of the initial step of taking the subsets of the special case $X=Y$. I wonder if we skip that special case and take the general case, can we extend the concept of partition and all these for the cartesian product of two or more sets as well? In that case, can we also claim that every partition of a set gives an equivalence relation and vice versa? I can clearly see it would be not that easy to define the relations but is there any work available that not necessarily use equivalence relation concept but have same essence?