By Ageev A.A.

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**Additional resources for A 3/2-approximation algorithm for the jump number of interval orders**

**Example text**

If I is complete, then (iv) Ve EEE: I'el = lel (v) Vc E CE: Icl = ·IBEI· t Proof (i) holds since I is simple. (ii) follows using the definition of a complement. 2. (iv) follows using the definition of a complement, since bE' e -= bEe'. (v) is implied by (iii). 0 Fig. 26. A condition b and it complement b (c) Definition, Let I be a C / E-system and let B s:; BE be the set of those conditions which have no complement in BE. For each b E B let b denote a new element. Let F= {(e, b) I (b, e) E FE /\ bE B} u {(h, e) I (e, b) E FE /\ bE B}.

Therefore (aj, a2) ~ FK . There must be at least one S-element s E L with aj < s < a2' By definition of aj, Y dE D : s co d. But this is impossible since D is a cut. 0 Figure 34 shows that unbounded occurrence nets are not always K-dense. 3 Processes We will now define processes of C/ E-systems using bounded occurrence nets. We will define this notion only for contact-free C/ E-systems, the reason for this will be discussed after having given the definition. 5 (i) (ii)). Processes will be described as mappings from bounded occurrence nets into contact-free C/E-systems satisfying two requirements: (i) Each slice is mapped injectively onto a case and (ii) the mapping of a T-element to an event respects the environment of the event.

1 takes account of the fact that, in a process, concurrency may yield slices which are not ordered. 1 (i) could otherwise not be obtained. We are now going to define a simpler function a', which is equivalent to the synchronic distance a in the special case of cyclic C/ E-systems. (a) Definition. Let L be a C/ E-system which is contact-free, let E I , E2 ~ E}; and let P E 7C};. We define v' (p, E I , E 2) = IIp- 1 (E I) I-Irl (E 2) I and a' (EI' E 2) = sup {v' (p, E I , E 2) Ip E 7C};}. (b) Proposition.