The paper writes Condition 1 as $\sum_{i \in Z,\ i \le j} p_i \le s_j$ with the jobs indexed in nondecreasing order of $s$. That phrasing pins the sum down only once ties are broken, and where the $s$ values are not distinct the form used here is the one the paper means: two jobs with the same $s$ must both have been preprocessed by that time, so both lengths count. A prefix of an enumeration that stopped at the first of them would not. Sections 4 and 6 arrange for the $s$ values to be distinct, and there the two readings agree; that they do is a statement of this submission rather than a convention of this file, and so is the fact that the reading below is equivalent to the existence of an actual schedule.

Condition 2 is written as a colouring rather than as a partition. The data are the same — a colour is the index of the part a job goes into — and a function is what the formalization of a schedule uses; that a partition can be recovered is immediate. The colour of a job outside $Z$ is unconstrained.

The sums are taken in $\mathbb{Z}$, where the start times live, so that no truncated subtraction appears in the condition.