By Raupp F.

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1 we have (a, b) E indo Consider ~ = acb = (ac,cb). In this case P0i/l. b(l). ' where ~ = (:C1,"" :cn ) can be obtained as a composition of total orders TO e1 , ... e. P0i/l. 2. e. 6 in Appendix A). Q. b(i) where i j = #b(yavb). 9 in Appendi:c A). Proof. 3. a(i». 4, (a,b) E ind, contradicting i(a) n i(b) '" 0. (3) Suppose i(a) n i(b) = 0. Clearly a '" b. ak(y)b) = j. Let Y = {y E lin(~)lk(y) ~ k(v) for all v E lin(~) } . Clearly for all y E Y, k(y) is the same so let us set k = k(y), y E Y. We have k ;:::: 1.

Eya·. We say that N = (S, T, F) is a T-restricted net if and only if it is a general. net and T = •S n S·. Throughout the rest of this book we shall only be concerned with T -restricted nets, which will simply be called nets. The condition T = •S n S· means that each transition has at least one pre-place and at least one postplace. For example the graphs of Fig. 4 (a), (b), (c) and Fig. 6 considered without tokens represent (T-restricted) nets. The two simple graphs from Fig. 7 represent general.

Nn } where Ni = (Si,Ti,Fi,Mt) fori = 1, .. ,n. Then: (1) (Vt E T)card(et) = cardW) = card({Nilt E Ti})' Chapter 2. Formal Theory of Basic COSY 52 {N1 ,N2 } is a state machine decomposition of N, and {Ng ,N4 } is also a state machine decomposition of N. Note that for instance {N1 ,N2 ,Ng ,N4 } is not a state machine decomposition of N because of (2) of the definition. 11. 2. 12. Contact free non smd-net (2) (VM E [Mo) )card(M) (3) N is contact-free. = n /\ (Vi = 1, ... ,n)card(M n Si) = 1. Proof.