I know a congruence (or congruence relation) is an equivalence relation on an algebraic structure, compatible with the operation on the structure. I am also familiar with equivalent definitions.
However, I am not able to find anything about what is a "discrete congruence". I came accross this term somewhere on stackexchange once and wonder how is it defined.
Can you help me explain what a discrete congruence is? My guess is that it is equivalence relation where every equivalence class has only one element? But it may be nonsense.
"The discrete congruence" on a structure $X$ (and also "the discrete equivalence relation" on a set $X$) refers to the equality relation $=$ on $X$: $$\{(a,a)\mid a\in X\}.$$ This is the minimal congruence / equivalence relation on $X$. Its equivalence classes are singletons.
In contrast, the maximal congruence / equivalence relation on $X$ is called "the trivial congruence" (or "the trivial equivalence relation"): $$\{(a,b)\mid a,b\in X\}.$$ It has a single equivalence class, $X$.
I believe the terminology comes from an analogy with the maximal and minimal topologies on a set: the discrete topology (in which singletons are open) and the trivial topology (in which the only non-empty open set is $X$). The terminology, while not universal, is at least somewhat common, as a google search reveals:
https://www.google.com/search?q=%22the+discrete+equivalence+relation%22
https://www.google.com/search?q=%22the+discrete+congruence%22