With precedence constraints an instance carries a relation on the jobs, given by its arcs, and a time lag $l_{ij}$ for every pair of jobs. Whenever $i$ precedes $j$, job $j$ may not start before the completion of job $i$ plus the lag: $C_i + l_{ij} \le S_j$. A lag may be negative.

A value of the field consists of a class of relations and a kind of lags.

The classes are, with $\prec$ the transitive closure of the arcs:

• none: no relation, read as the empty relation;
• chains: every job has at most one successor and at most one predecessor;
• intree: every job has at most one successor;
• outtree: every job has at most one predecessor;
• tree: the relation is an intree or an outtree;
• opposing forest: the jobs split into two parts without arcs between them, one an intree and the other an outtree;
• series-parallel: $\prec$ contains no four jobs $a, b, c, d$ whose only relations are $a \prec c$, $b \prec c$, $b \prec d$;
• divide-and-conquer graph: a single job is one; and for two further jobs $s$ and $t$ and at least one divide-and-conquer graph on disjoint sets of jobs, so is their union with $s$, $t$ and all arcs from $s$ to the jobs of the graphs and from these to $t$;
• interval order: $a \prec b$ and $c \prec d$ imply $a \prec d$ or $c \prec b$;
• over-interval order: no five different jobs $1, \dots, 5$ are related exactly by $1 \prec 2$, $3 \prec 4$, $3 \prec 5$, and none exactly by the reverse relations $2 \prec 1$, $4 \prec 3$, $5 \prec 3$ (the structures of type I and II of Chardon and Moukrim, 2005);
• quasi-interval order: an over-interval order in which, moreover, no five different jobs are related exactly by $1 \prec 2$, $1 \prec 4$, $3 \prec 5 \prec 4$ and hence $3 \prec 4$ (Moukrim 1999, Definition 2: the structures of type I, in either orientation, and of type II);
• $A_m$-order, for the number $m$ of machines of the instance: for any two sets $A$ and $B$ of at most $m$ pairwise unrelated jobs, at most $m$ jobs of $A \cup B$ have a predecessor in $A \cup B$, and at most $m$ have a successor in $A \cup B$ (the property $Q_m$ of Moukrim and Quilliot, 1997);
• level order: every job of level $l + 1$ precedes every job of level $l$, where the level of a job is the length of the longest path of arcs starting from it;
• bounded height: an arbitrary relation; the value records that the length of its longest path is regarded as a constant;
• dimension two: the jobs can be placed in the plane such that, for different jobs, $i \prec j$ exactly if both coordinates of $i$ are at most those of $j$;
• bounded dimension: the same in a space of some dimension;
• prec: an arbitrary relation.

The class none is a special case of every class except divide-and-conquer graphs, which have a single first and a single last job; interval orders of quasi-interval orders, of over-interval orders and of $A_m$-orders; quasi-interval orders of over-interval orders; chains of intree and outtree; these of tree; tree of opposing forest; dimension two of bounded dimension; chains, the trees and opposing forests of series-parallel orders, since every job of an intree has its successors on a chain and every job of an outtree its predecessors; series-parallel orders of dimension two and of bounded dimension, by the classical theorem that an order without such four jobs has dimension at most two; and every class of prec.

The kinds of lags are: zero; unit, $l_{ij} = 1$; constant, $l_{ij} = l$, also restricted to $l \le 0$ or to $l > 0$; and arbitrary, also restricted to $l_{ij} \le 0$ or to $l_{ij} > 0$. A kind is a special case of another if its lags are among the other's.

On a single machine, as in batching, every order is an $A_1$-order, so the value says nothing there. The field of one machine leaves it out: the value $A_m$ admits no instance and is a special case of no other value.

Theorem. Over every kernel, the field of precedence constraints with time lags is sound, and so is the field of one machine.