Left congruence

''The notion of left congruence also makes sense in the more general context of a monoid. In fact, the same definition works.''

Symbol-free definition
A left congruence on a group is an equivalence relation on the group with the property that the equivalence relation is preserved on left multiplication by any element of the group.

Definition with symbols
A left congruence on a group $$G$$ is an equivalence relation $$\equiv$$ on $$G$$ such that:

$$a \equiv b \implies ca \equiv cb$$

Relation with other notions
The dual notion to left congruence is the notion of right congruence.

An equivalence relation is termed a congruence if it is both a left congruence and a right congruence.

Correspondence between subgroups and left congruences
The following is true:

Left congruences are precisely the equivalence relations whose equivalence classes are the left cosets of a subgroup

Proving that any left congruence gives left cosets
We first show that the equivalence class of the identity element is a subgroup. For this, we show the following three things:


 * Identity elements:The identity element is equivalent to the identity element: This follows on account of the relation being reflexive
 * Closure under multiplication: If $$a,b \equiv e$$, so is $$ab$$: The proof of this comes as follows. Suppose $$b \equiv e$$. Then $$ab \equiv a$$. We already know that $$a \equiv e$$. Hence, by the transitivity of $$\equiv$$, we have $$ab \equiv e$$.
 * Closure under inverses: If $$a \equiv e$$, then we can pre-multiply both sides by $$a^{-1}$$ and obtain $$e \equiv a^{-1}$$

Let $$H$$ denote this subgroup. Then clearly, for any $$x \in G$$, $$x \equiv xh$$ (left multiplying $$e \equiv h$$ by $$x$$). Thus all the elements in the left coset of $$H$$ are in the same equivalence class as $$x$$.

Further, we can show that if $$x \equiv y$$, they must be in the same left coset. Suppose $$x \equiv y$$. Then, left multpily both sides by $$y^{-1}$$. This gives $$y^{-1}x \equiv e$$, hence $$y^{-1}x \in H$$ or $$y \in xH$$.

Proving that left cosets give a left congruence
This is more or less direct.