I am familiar with this (informal) definition of congruence relation (or simply congruence):
(1) A congruence is an equivalence relation on an algebraic structure that is compatible with the structure in the sense that algebraic operations done with equivalent elements will yield equivalent elements.
(Particular definition of a congruence then depends on the type of algebraic structure under consideration).
Wikipedia says that
An equivalent formulation in the universal algebra context is the following:
(2) A congruence relation on an algebra A is a subset of the direct product A × A that is both an equivalence relation on A and a subalgebra of A × A.
My question: Is the definition (2) valid for any algebraic structure? And intuitively, why are these two definitions ((1),(2)) compatible?
I mean, it is clear to me that any congruence relation is an equivalence relation. But the second criterium is just being a subalgebra of the direct product, and here I feel like I don´t see why it is enough to see that we have a congruence. How does being a subalgebra of A x A "contribute" to being compatible with the algebraic structure?
I hope my question is clear, thank you for providing any intuition.
Yes, (2) is valid for all universal algebraic structures.
A relation $R\subseteq A\times B$ being a subalgebra expresses exactly that $R$ is compatible with the operations of $A$ and $B$:
$$(a_i,b_i)\in R\implies \underset i{\large\mu\,}(a_i,b_i)=(\underset i{\large\mu\,}a_i,\ \underset i{\large\mu\,} b_i)\in R$$ for any basic operation $\mu$.