By Jørn Børling Olsson

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8 now show that the (p, q)-string containing I consists solely of I1 and I2 , as described in case (iii) (b). Similarly, if I is involved in a chain of the form I1 ≺q I2 but not in one of the form I1 ≺p I2 , then we are in case (iii) (c). The other possibility is that I is involved in a chain of the form J1 ≺p J2 and in a chain of the form J3 ≺q J4 . There are four subcases to consider: I1 ≺p I ≺q I4 , I1 ≺q I ≺p I4 , I3 q I ≺p I2 and I3 ≺p I q I2 . 10. 8, we find that the (p, q)-string containing I consists only of these four ideals, and this completes the proof of (iii).

The support of a heap ε : E → is the subgraph of whose vertices are ε(E). 1 shows a heap E of size 5 over a graph with three vertices. In this case, the labelling function ε : E → satisfies ε(a) = ε(d) = 1, ε(c) = 2 and ε(b) = ε(e) = 3. The support of E is the whole of . The vertex chains of E are ε−1 (1) = {a, d}, ε −1 (2) = {c} and ε−1 (3) = {b, e}. The edge chains of E are ε −1 ({1, 2}) = {a, c, d} and ε −1 ({2, 3}) = {b, c, e}. The dual heap, E ∗ , has the same underlying set and labelling function, but the relations d < c < a and e < c < b in E become a <∗ c <∗ d and b <∗ c <∗ e.

Part (ii) is a consequence of (i). If x and y are comparable, then we may assume without loss of generality that x ≤ y. The sequence ε(z0 ), . . , ε(zk ) of (i) then produces a path in from ε(x) to ε(y), which implies that ε(x) and ε(y) lie in the same connected component of . If y covers x in E as in (iii), we must have k = 1 in the sequence of (i), and the assertion follows. If E is locally finite, the sequence in (i) may be refined if necessary until the relations shown are covering relations.

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