# Is there a name for an relation that is in some sense “compatible” with an operation.

By "compatible" I mean, if $R$ is my relation and $+$ my operation, then if $aRb$ and $cRd$, then $(a+c)R(b+d)$. For an congruence relation this is claimed, but also that $R$ must be an equivalence relation, but there are relations, like $<$ that fulfill the first statement, but are not equivalence relation's. Are there any terms for such relations and are they studied in mathematics?

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You have chosen the right word, compatible is a standard technical term for the notion. – André Nicolas Feb 1 '13 at 19:33

Considering a functional relation $R$, the compatability says, in functional notation: $R(a+c)=R(a)+R(c)$. In other words, the function preserves that operation. Of course we study examples of such relations all the time whenever we are looking at homomorphisms!
He talks about regarding a homomorphism of things in a category with an operation $S\rightarrow B/L$ instead as a "many valued function" from $S$ into $B$ (that is, every pre-image is related to the multiple things in the coset of the image). It remains "compatible", but is no longer a true function.
Another way to look at R preserves the operation is that it commutes with it. But is this the same setup as the OP's? $aRb \wedge cRd \to (a+c)R(b+d)$ versus $R(a+b)=R(a)+R(b)$? There are obvious distinctions such as the number of arguments, implication versus equality, and also in the first case there is a conjunction and $+$, while in the latter only $+$ operator. – alancalvitti Feb 2 '13 at 2:13