0. The concept

Let's start with a familiar case: congruence mod $n$ on the ring Z of integers. Just to be specific, let's use $n=6$. This congruence is an equivalence relation that is compatible with the ring operations, in the following sense:

	$a$	$\equiv$	$b$
	$a'$	$\equiv$	$b'$
$\Rightarrow$	$a+a'$	$\equiv$	$b+b'$

	$a$	$\equiv$	$b$
	$a'$	$\equiv$	$b'$
$\Rightarrow$	$aa'$	$\equiv$	$bb'$

	$a$	$\equiv$	$b$
$\Rightarrow$	$-a$	$\equiv$	$-b$

and of course $0 \equiv 0$.

The same definition works for algebraic systems in general:

0.1 Definition. A congruence relation on an algebra ${\cal A} = \langle A; f _ 1,\dots, f _ m \rangle$ is an equivalence relation $\equiv$ that is compatible with the operations, in the sense that for each basis operation $f _ i$, if $f _ i$ is $n _ i$-ary we have

$a _ 1 \equiv b _ 1$, ..., $a _ {n _ i} \equiv b _ {n _ i} \Rightarrow f _ i(a _ 1,\dots, a _ {n _ i}) \equiv f _ i(b _ 1,\dots, b _ {n _ i})$.

Terminology. Often we name a congruence relation $\theta$, say, and write either $a \theta b$ or $a \equiv b\; (\theta)$. Also, we may say ``congruence'' instead of ``congruence relation''. Just as for equivalence relations in general, we can speak of the blocks of a congruence relation (or ``classes'', but that usage is somewhat old). For $a \in A$, the block of $a$ is often called $\bar a$.