What are subsets of $A\times B$ called in mathematics?
In logic (algebra and analysis), a binary relation on/over $A$ is a subset of $A^2$. Abstractions of $=$, $\leq$, etc. While using subsets of $A\times B$ extensively in the role of functor, logicians have no term for these objects. I have always called a subset of $A\times B$ a binary relationship, but always explain that this is just my term.
In the computer science ER Model, a binary relation is a subset of $A\times B$, with a subset of $A^2$ being called a recursive binary relation.
Wikipedia currently makes a complete mess of the situation, by defining a binary relation as
In mathematics, a binary relation over two sets $X$ and $Y$ is a set of ordered pairs $(x, y)$ consisting of elements $x\in X$ and $y\in Y$. That is, it is a subset of the Cartesian product $X × Y$.
yet defining an equivalence relation, via reference to the above definition, as
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive.
The latter definition is complete nonsense for subsets of $A\times B$, only making sense for subsets of $A^2$. Reflexivity cannot be defined for subsets of $A\times B$.
- What are subsets of $𝐴×𝐵$ called in mathematics?
- I have never encountered these being called binary relations in mathematics, except for Wikipedia. I thought this might be my algebra-logic bias, but have checked Analysis books, and yes, (binary) relations are as we have known them from primary school, subsets of $A^2$. Abstraction of $=$, $\leq$, etc.
- Unary operations ($o:A\rightarrow A$) are to binary relations ($r\subseteq A^2$) as functions ($f:A\rightarrow B$) are to ??? ($R\subseteq A\times B$)?
I am not looking for a proposed term, nor asking for reference back to Wikipedia, rather, I am asking if there is some term already in widespread use in the literature, and ideally with reference.
Based on answers received so far, an additional question would then be,
- in the context that binary relations are subsets of $A\times B$, what term distinguishes the $A^2$ case? Endomorphic? Recursive?
Some have argued that the Wikipedia definition of equivalence relation is correct as stands. It is not. The definition of an equivalence relation cannot be phrased for subsets of $A\times B$. Reflexivity is impossible to define for subsets of $A\times B$. Further, no one would define an equivalence relation as a subset of $A\times B$ such that..., and then expect the reader to deduce that $A$ must equal $B$. No. That is not how equivalence relations are defined. Currently Wikipedia is wrong.
@TheEmptyFunction has drawn my attention to the fact that Wikipedia currently defines an $n$-ary operation as a function $A_1\times...\times A_n\rightarrow B$, rather than the mathematicians $A^n\rightarrow A$. This definition strips the notion of operation of all its logical/algebraic/analytic meaning. Homomorphisms no longer make sense. The same is true for calling a subset of $A\times B$ a binary relation.