Suppose I have a set of 3-tuples $A = \{(3, 4, 5), (3, 4, 6), (2, 5, 7), (1, 1, 1)\}$.
I want to define an ordering over elements in this set so that for two elements $x=(x_1, x_2, x_3)$ and $y=(y_1, y_2, y_3)$, $x<y$ if either $(x_1 < y_1)$ or $(x_1 = y_1) \land (x_2 < y_2)$.
If $(x_1 = y_1) \land (x_2 = y_2)$, I want to have the ordering break ties uniformly at random.
I think the notation for the first part is clear enough (correct me if I'm wrong or if I'm using bad notation/terminology), but it's not clear to me how to more formally describe the "randomly breaking ties" part. Any suggestions?