Reflexive Euclidean Relation is Equivalence

Theorem
A relation is an equivalence iff it is either left-Euclidean or right-Euclidean, and also reflexive.

Sufficient Condition
Let $$\mathcal{R}$$ be an equivalence relation.

Then by definition it is reflexive.

It is also transitive and symmetric.

So, let $$x, y, z \in \mathcal{R}$$ such that $$x \mathcal{R} y$$ and $$x \mathcal{R} z$$.

From symmetry, we have $$y \mathcal{R} x$$, and by transitivity it follows that $$y \mathcal{R} z$$.

Hence $$\mathcal{R}$$ is right-Euclidean.

Now let $$x, y, z \in \mathcal{R}$$ such that $$x \mathcal{R} z$$ and $$y \mathcal{R} z$$.

From symmetry, we have $$z \mathcal{R} y$$, and by transitivity it follows that $$x \mathcal{R} y$$.

Hence $$\mathcal{R}$$ is left-Euclidean.

So $$\mathcal{R}$$ is both left-Euclidean and right-Euclidean.

We know by definition that it is reflexive.

Thus an equivalence relation is both left-Euclidean and right-Euclidean and also reflexive.

Necessary Condition
Let $$\mathcal{R}$$ be a relation which is both left-Euclidean and reflexive.

Checking in turn each of the criteria for equivalence:

Reflexive
By definition: $$\mathcal{R}$$ has been defined as a relation which is reflexive.

Symmetric
Suppose $$x \mathcal{R} y$$.

We have that $$y \mathcal{R} y$$ from the reflexivity of $$\mathcal{R}$$.

So we have $$y \mathcal{R} y$$ and $$x \mathcal{R} y$$, and so $$y \mathcal{R} x$$ from the fact that $$\mathcal{R}$$ is left-Euclidean.

Hence $$\mathcal{R}$$ is symmetric.

Transitive
Suppose $$x \mathcal{R} y$$ and $$y \mathcal{R} z$$.

As $$\mathcal{R}$$ is symmetric (from above), we have that $$z \mathcal{R} y$$.

So we have $$x \mathcal{R} y$$ and $$z \mathcal{R} y$$, and by the fact that $$\mathcal{R}$$ is left-Euclidean, we have that $$x \mathcal{R} z$$.

So, if $$\mathcal{R}$$ be a relation which is both left-Euclidean and reflexive, it is also an equivalence relation.

A similar argument shows that if $$\mathcal{R}$$ be a relation which is both right-Euclidean and reflexive, it is also an equivalence relation.