Remark 2

The conjunction of axioms (U_{II}) and (U_{III}) is equivalent (assuming the other axioms of uniform structures) to the following axiom:

(U_a) For each $V \in \mathfrak{U}$ there exists $W \in \mathfrak{U}$ such that $W \circ \overline{W} \subset V$ (*).

Clearly (U_{II}) and (U_{III}) imply (U_a). Conversely, if (U_a) is satisfied we have $\overline{W} = \Delta \circ W \subset V$, by (U_I); hence $W \subset \overline{V}$ and therefore [by (F_1)] $\overline{V} \in \mathfrak{U}$. Let $W' = W \cap \overline{W}$; then $W' \in \mathfrak{U}$ by what has just been proved and axiom (F_{II}), and we have $W' \circ W' \subset W \circ \overline{W} \subset V$.

Throughout this chapter we shall write $\overline{V}^2$ instead of $V \circ V$, and in general $\overline{V}^{n-1} = \overline{V} \circ V = V \circ \overline{V}^{n-1}$, for each integer $n > 1$ and each subset $V$ of $X \times X$.