The definition of two-sided uniformity

Question

In the field of uniform spaces, what does it mean to be a two-sided uniformity? Could not find a clear definition of this online.

Answer 1

According to Definition $2.2$ in [1]:

Let $(X,\tau)$ be a topological group. The supremum $\mathfrak{L}\vee \mathfrak{R}$ of the left and right uniformities $\mathfrak{L}$ and $\mathfrak{R}$ of $(X,\tau)$ will be called the upper uniformity or, more traditionally, the two-sided uniformity of $(X,\tau)$.

The basis for this uniformity is formed by the sets $$\{(x,y)\in X\times X: y\in xU\cap Ux\}$$ where $U$ is a neighborhood of the identity.

[1] W. Roelcke, S. Dierolf, Uniform Structures on Topological Groups and their Quotients (1981).