3. Algebraic operators COMP and SYNC and their properties

We study the issue of transforming PNTS through precisely defined binary operator COMP and n-ary operator SYNC over the class PNTS and we also examine the preservation of individual PNTS's properties when applying each of these operators. Formal enrollment of an application of generally n-ary operator OP whose operands are the PNTS PNTS1, PNTS2, …, PNTSn (n ∈ N) and whose application requires the specification of values of k formal parameters (k ∈ N) par1, par2, … park, will be denoted by the expression.

> PNTS≔½ � PNTS1; PNTS2; …; PNTSn :OP par1; par2;…; park ,

where PNTS is the resulting PNTS.

ASe ∈ ASe such that the output state Sx is also reachable from the state S (i.e., ∀S ∈ [ASe〉 ∃Sx ∈ [ASe〉x: Sx ∈ [S〉). Furthermore, the cardinality of the set [ASe〉〉 of all the sequences r := σ<sup>1</sup> δ<sup>1</sup> σ<sup>2</sup> δ<sup>2</sup> … σ<sup>n</sup> δ<sup>n</sup> associated with all the reachable states S ∈ [ASe〉〉 must be finite (i.e., (∃n ∈ N: |

Proper-formed PNTS is well-formed PNTS if for any of its allowed entry state ASe ∈ ASe and for any of its exit state Sx ∈ [ASe〉x, where Sx := (Mx, mx, τx), it is true that the exit static marking ξ(Mx) of all its resource places is an element of the set AMs of all its allowed static markings if PNTS PNTS be in its allowed entry state ASe ∈ ASe (i.e., ∀ASe ∈ ASe ∀Sx ∈ [ASe〉x, Sx := (Mx, mx,

Well-formed PNTS is pure-formed PNTS if for any of its allowed entry state ASe ∈ ASe, where ASe := (AMe, ame, τe), and for any of its exit state Sx ∈ [ASe〉x, where Sx := (Mx, mx, τx), it is true that the exit static marking ξ(Mx) of all its resource places is equal to the entry static marking ξ(AMe) of all its resource places that is associated with the allowed entry state ASe (i.e., ∀ASe ∈

i. AMs := {(0, 0, k, 0) | k ∈ N} (i.e., there must be at least one token in the resource place R1 in any allowed entry state ASe ∈ ASe), then it can be shown that PNTS PNTS1 is k-bounded,

ii. AMs := {(0, 0, 0, 0)} (i.e., there may not be any token in the resource place R1 in any allowed entry state ASe ∈ ASe), then it can be shown that PNTS PNTS1 is k-bounded, proper-formed, but not well-formed or pure-formed PNTS (see for instance the sequence

ð Þ ð Þ 1; 0; 0; 0 ; <ð Þ 0 >; <>; <>; <> ; 0 ½T2i ð Þ ð Þ 0; 2; 0; 0 ; <>; < ð Þ 3; 3 >; <>; <> ; 0 ½3i: ð Þ ð Þ 0; 2; 0; 0 ; <>; < ð Þ 3; 3 >; <>; <> ; 3 ½T3 T3i ð Þ ð Þ 0; 0; 2; 2 ; <>; <>; < ð Þ 4; 4 >; < 7; 7 > ; 3 ,

Proof. Clear. PNTS PNTS := (P, T, A, AF, TP, TI, IP, OP, RP) is a connected net that contains the finite set T of the transitions. Then the finite number of tokens will be added to each of the places p ∈ P by firing each of the transitions t ∈ T. The number of states S ∈ [ASe〉 for any allowed entry state ASe ∈ ASe must be also finite because PNTS is proper-formed PNTS (i.e., ∃n ∈ N: |[ASe〉〉| = n). From these facts then immediately follows that in any state S ∈ [ASe〉 the finite number of tokens must be placed in any place p ∈ P, where any final number of tokens is placed in the input place IP in the entry state ASe. From these facts then immediately follows that <sup>∀</sup>ASe <sup>∈</sup> ASe <sup>∃</sup><sup>k</sup> <sup>∈</sup> <sup>N</sup><sup>0</sup> <sup>∀</sup>p<sup>∈</sup> <sup>P</sup> <sup>∀</sup>S<sup>∈</sup> ASe ½ i, S≔ð Þ <sup>M</sup>; <sup>m</sup>; <sup>τ</sup> : M pð Þ <sup>≤</sup> <sup>k</sup>: □

Definition 6. Process Petri net with time stamps (PPNTS) PPNTS is an ordered couple PPNTS := (PNTS, Se), where PNTS := (P, T, A, AF, TP, TI, IP, OP, RP) is the PNTS and Se ∈ Se is the entry state of PNTS PNTS. The class of all PPNTSs will be denoted by PPNTS. □

[ASe〉〉| = n).

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τx): ξ(Mx) ∈ AMs).

ASe ∀Sx ∈ [ASe〉x, Sx := (Mx, mx, τx): ξ(AMe) = ξ(Mx)).

For instance, if the set AMs of the PNTS PNTS1 (see Figure 1) is defined as:

where ξð Þ¼ Mx ξð Þ¼ ð Þ 0; 0; 2; 2 ð Þ 0; 0; 2; 0 ∉f g ð Þ 0; 0; 0; 0 ¼ AMsÞ:

Lemma 1. If PNTS is proper-formed PTNS then PNTS is k-bounded PNTS.

proper-formed, well-formed and pure-formed PNTS,

Definition 7. Let PNTS<sup>1</sup> := (P1, T1, A1, AF1, TP1, TI1, IP1, OP1, RP1) and PNTS<sup>2</sup> := (P2, T2, A2, AF2, TP2, TI2, IP2, OP2, RP2) be the PNTSs. Let AMs<sup>1</sup> := {(AMs1(IP1), AMs1(p11), …, AMs1(p1n), AMs1(r11), …, AMs1(r1m), AMs1(OP1)) | P<sup>1</sup> := {p11, …, p1n, r11, …, r1m}, RP<sup>1</sup> := {r11, …, r1m}, n ∈ N, m ∈ N} be the set of all the allowed static markings of PNTS1, AMs<sup>2</sup> := {(AMs2(IP2), AMs2(p21), …, AMs2(p2k), AMs2(r21), …, AMs2(r2h), AMs2(OP2)) | P<sup>2</sup> := {p21, …, p2k, r21, …, r2h}, RP<sup>2</sup> := {r21, …, r2h}, k ∈ N, h ∈ N} be the set of all the allowed static markings of PNTS2.

Cartesian product AMs<sup>1</sup> ⊗ AMs<sup>2</sup> is then the following set:

$$\begin{split} \mathbf{AM}\_{\mathfrak{sl}1} \otimes \mathbf{AM}\_{\mathfrak{sl}2} &= \{ (\mathbf{AM}\_{\mathfrak{sl}1}(\mathbf{l}\mathbf{1}\_{1}), \mathbf{AM}\_{\mathfrak{sl}1}(\mathfrak{p}\mathbf{1}\_{1}), \dots, \mathbf{AM}\_{\mathfrak{sl}1}(\mathfrak{p}\mathbf{1}\_{n}), \mathbf{AM}\_{\mathfrak{sl}1}(\mathfrak{r}\mathbf{1}\_{1}), \dots, \mathbf{AM}\_{\mathfrak{sl}1}(\mathfrak{r}\mathbf{1}\_{m}), \mathbf{AM}\_{\mathfrak{sl}1}(\mathbf{OP}\_{1}), \\ &\quad \text{.} \quad \text{.} \quad \text{.} \quad \text{.} \quad \text{.} \quad \text{.} \quad \text{.} \quad \text{.} \quad \text{.} \quad \text{.} \quad \text{.} \quad \text{.} \quad \text{.} \quad \text{.} \quad \text{.} \quad \text{.} \quad \text{.} \quad \text{.} \quad \text{.} \quad \text{.} \quad \text{.} \quad \text{.} \quad \text{.} \quad \text{.} \quad \text{.} \quad \text{.} \quad \text{.} \quad \text{.} \quad \text{.} \quad \text{.} \quad \text{.} \quad \text{.} \quad \text{.} \quad \text{.} \quad \text{.} \quad \text{.} \quad \text{.} \quad \text{.} \quad \text{.} \quad \text{.} \quad \text{.} \quad \text{.} \quad \text{.} \quad \text{.} \quad \text{.} \quad \text{.} \quad \text{.} \quad \text{.} \quad \text{.} \quad \text{.} \quad \text{.} \quad \text{.} \quad \text{.} \quad \text{.} \quad \text{.} \quad \text{.} \quad \text{.$$

PNTS PNTS<sup>1</sup> and PNTS<sup>2</sup> are disjoint and we denote this fact by PNTS<sup>1</sup> ∠ PNTS<sup>2</sup> if.

$$(P\_1 \cap P\_2 = \mathcal{Q}) \land (T\_1 \cap T\_2 = \mathcal{Q}).\tag{7}$$

Definition 8. The function COMP: PNTS � PNTS ! PNTS of nets composition is defined as follows: if PNTS<sup>1</sup> := (P1, T1, A1, AF1, TP1, TI1, IP1, OP1, RP1) and PNTS<sup>2</sup> := (P2, T2, A2, AF2, TP2, TI2, IP2, OP2, RP2) be the arbitrary PNTSs, PNTS<sup>1</sup> ∠ PNTS2, t be an arbitrary transition, where (t ∉ T1) ∧ (t ∉ T2), ti ∈ N0, then PNTS := [PNTS1, PNTS2].COMP(t, ti), where PNTS PNTS := (P, T, A, AF, TP, TI, IP, OP, RP) fulfills the following:


$$\mathbf{\dot{x}}.\qquad RP \coloneqq RP\_1 \cup RP\_2.$$

If PNTS<sup>1</sup> and PNTS<sup>2</sup> are proper-formed, resp. well-formed, resp. pure-formed, PNTS and AMs = AMs<sup>1</sup> ⊗ AMs<sup>2</sup> be the set of all the allowed static markings of PNTS PNTS, then also

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Proof. Clear, it directly follows from Definition 5, Definition 7 and Definition 8. □

Definition 9. The function SYNC: PNTS � PNTS � … � PNTS ! PNTS of synchronous nets composition is defined as follows: if PNTS<sup>1</sup> := (P1, T1, A1, AF1, TP1, TI1, IP1, OP1, RP1), PNTS<sup>2</sup> := (P2, T2, A2, AF2, TP2, TI2, IP2, OP2, RP2), …, PNTSn := (Pn, Tn, An, AFn, TPn, TIn, IPn, OPn, RPn), be the arbitrary PNTSs, ∀i, 1 ≤ i ≤ n, ∀j, 1 ≤ j ≤ n: i 6¼ j ) PNTSi ∠ PNTSj, where n ∈ N, pi and po be the arbitrary places, (pi ∉ P<sup>1</sup> ∪ P<sup>2</sup> ∪ … ∪ Pn) ∧ (po ∉ P<sup>1</sup> ∪ P<sup>2</sup> ∪ … ∪ Pn) ∧ (pi 6¼ po), ti and to be the arbitrary transitions, (ti ∉ T<sup>1</sup> ∪ T<sup>2</sup> ∪ … ∪ Tn) ∧ (to ∉ T<sup>1</sup> ∪ T<sup>2</sup> ∪ … ∪ Tn) ∧ (ti 6¼ to), af1 ∈ N, af2 ∈

PNTS≔½ � PNTS1; PNTS2;…; PNTSn :SYNCð Þ pi; po; ti; to; af 1;…; afn; ti1;…; tin; tio ,

iii. A := A<sup>1</sup> ∪ A<sup>2</sup> ∪ … ∪ A<sup>n</sup> ∪ {(pi, ti), (ti, IP1), …, (ti, IPn), (OP1, to), …, (OPn, to), (to, po)},

iv. AF := AF<sup>1</sup> ∪ AF<sup>2</sup> ∪ … ∪ AF<sup>n</sup> ∪ {((pi, ti), 1), ((ti, IP1), af1), …, ((ti, IPn), afn), ((OP1, to), af1), …,

ix. RP := RP<sup>1</sup> <sup>∪</sup> RP<sup>2</sup> <sup>∪</sup> … <sup>∪</sup> RPn. □ Symbolic representation of PNTS [PNTS1, PNTS2, …, PNTSn].SYNC(pi, po, ti, to, af1, …, afn, ti1,

Lemma 3. Let PNTS<sup>1</sup> := (P1, T1, A1, AF1, TP1, TI1, IP1, OP1, RP1), PNTS<sup>2</sup> := (P2, T2, A2, AF2, TP2, TI2, IP2, OP2, RP2), …, PNTSn := (Pn, Tn, An, AFn, TPn, TIn, IPn, OPn, RPn) be arbitrary PNTSs, ∀i, 1 ≤ i ≤ n, ∀j, 1 ≤ j ≤ n: i 6¼ j ) PNTSi ∠ PNTSj, where n ∈ N, pi and po be arbitrary places, (pi ∉ P<sup>1</sup> ∪ P<sup>2</sup> ∪ … ∪ Pn) ∧ (po ∉ P<sup>1</sup> ∪ P<sup>2</sup> ∪ … ∪ Pn) ∧ (pi 6¼ po), ti and to be arbitrary transitions, (ti ∉ T<sup>1</sup> ∪ T<sup>2</sup> ∪ … ∪ Tn) ∧ (to ∉ T<sup>1</sup> ∪ T<sup>2</sup> ∪ … ∪ Tn) ∧ (ti 6¼ to), af1 ∈ N, af2 ∈ N, …, afn ∈ N, ti1 ∈ N0, ti2 ∈ N0, …, tin ∈ N0, tio ∈ N<sup>0</sup> and AMs1, AMs2, …, AMs<sup>n</sup> be the sets of all the allowed static markings of PNTS1, PNTS2, …, PNTSn. Let PNTS := [PNTS1, PNTS2, …, PNTSn].SYNC(pi, po, ti, to, af1, …,

resulting PNTS is proper-formed, resp. well-formed, resp. pure-formed, PNTS.

N, …, afn ∈ N, ti1 ∈ N0, ti2 ∈ N0, …, tin ∈ N0, tio ∈ N0, then

i. P := P<sup>1</sup> ∪ P<sup>2</sup> ∪ … ∪ P<sup>n</sup> ∪ {pi, po}, ii. T := T<sup>1</sup> ∪ T<sup>2</sup> ∪ … ∪ T<sup>n</sup> ∪ {ti, to},

((OPn, to), afn), ((to, po), 1)},

…, tin, tio) can be seen in Figure 3.

vii. IP := pi, viii. OP := po,

afn, ti1, …, tin, tio).

v. TP := TP<sup>1</sup> ∪ TP<sup>2</sup> ∪ … ∪ TP<sup>n</sup> ∪ {(ti, 1), (to, 1)},

where PNTS PNTS := (P, T, A, AF, TP, TI, IP, OP, RP) fulfills the following:

vi. TI := TI<sup>1</sup> ∪ TI<sup>2</sup> ∪ … ∪ TI<sup>n</sup> ∪ {((ti, IP1), ti1), …, ((ti, IPn), tin), ((to, po), tio)},

Symbolic representation of PNTS [PNTS1, PNTS2].COMP(t, ti) can be seen in Figure 2.

Lemma 2. Let PNTS<sup>1</sup> := (P1, T1, A1, AF1, TP1, TI1, IP1, OP1, RP1) and PNTS<sup>2</sup> := (P2, T2, A2, AF2, TP2, TI2, IP2, OP2, RP2) be two arbitrary PNTS, PNTS<sup>1</sup> ∠ PNTS2, t be an arbitrary transition, (t ∉ T1) ∧ (t ∉ T2), ti ∈ N0, AMs<sup>1</sup> and AMs<sup>2</sup> be the sets of all the allowed static markings of PNTS<sup>1</sup> and PNTS2. Let PNTS := [PNTS1, PNTS2].COMP(t, ti).

Figure 2. Symbolic representation of PNTS [PNTS1, PNTS2].COMP(t, ti).

If PNTS<sup>1</sup> and PNTS<sup>2</sup> are proper-formed, resp. well-formed, resp. pure-formed, PNTS and AMs = AMs<sup>1</sup> ⊗ AMs<sup>2</sup> be the set of all the allowed static markings of PNTS PNTS, then also resulting PNTS is proper-formed, resp. well-formed, resp. pure-formed, PNTS.

Proof. Clear, it directly follows from Definition 5, Definition 7 and Definition 8. □

Definition 9. The function SYNC: PNTS � PNTS � … � PNTS ! PNTS of synchronous nets composition is defined as follows: if PNTS<sup>1</sup> := (P1, T1, A1, AF1, TP1, TI1, IP1, OP1, RP1), PNTS<sup>2</sup> := (P2, T2, A2, AF2, TP2, TI2, IP2, OP2, RP2), …, PNTSn := (Pn, Tn, An, AFn, TPn, TIn, IPn, OPn, RPn), be the arbitrary PNTSs, ∀i, 1 ≤ i ≤ n, ∀j, 1 ≤ j ≤ n: i 6¼ j ) PNTSi ∠ PNTSj, where n ∈ N, pi and po be the arbitrary places, (pi ∉ P<sup>1</sup> ∪ P<sup>2</sup> ∪ … ∪ Pn) ∧ (po ∉ P<sup>1</sup> ∪ P<sup>2</sup> ∪ … ∪ Pn) ∧ (pi 6¼ po), ti and to be the arbitrary transitions, (ti ∉ T<sup>1</sup> ∪ T<sup>2</sup> ∪ … ∪ Tn) ∧ (to ∉ T<sup>1</sup> ∪ T<sup>2</sup> ∪ … ∪ Tn) ∧ (ti 6¼ to), af1 ∈ N, af2 ∈ N, …, afn ∈ N, ti1 ∈ N0, ti2 ∈ N0, …, tin ∈ N0, tio ∈ N0, then

PNTS≔½ � PNTS1; PNTS2;…; PNTSn :SYNCð Þ pi; po; ti; to; af 1;…; afn; ti1;…; tin; tio ,

where PNTS PNTS := (P, T, A, AF, TP, TI, IP, OP, RP) fulfills the following:


$$\mathbf{v}\mathbf{.} \qquad \mathbf{TP} := \mathbf{T}\mathbf{P}\_1 \cup \mathbf{T}\mathbf{P}\_2 \cup \dots \cup \mathbf{T}\mathbf{P}\_n \cup \{ (ti, 1), (to, 1) \} \dots$$


v. TP := TP<sup>1</sup> ∪ TP<sup>2</sup> ∪ {(t, 1)},

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vii. IP := IP1, viii. OP := OP2,

vi. TI := TI<sup>1</sup> ∪ TI<sup>2</sup> ∪ {(t, IP2), ti)},

and PNTS2. Let PNTS := [PNTS1, PNTS2].COMP(t, ti).

Figure 2. Symbolic representation of PNTS [PNTS1, PNTS2].COMP(t, ti).

ix. RP := RP<sup>1</sup> <sup>∪</sup> RP2. □

Lemma 2. Let PNTS<sup>1</sup> := (P1, T1, A1, AF1, TP1, TI1, IP1, OP1, RP1) and PNTS<sup>2</sup> := (P2, T2, A2, AF2, TP2, TI2, IP2, OP2, RP2) be two arbitrary PNTS, PNTS<sup>1</sup> ∠ PNTS2, t be an arbitrary transition, (t ∉ T1) ∧ (t ∉ T2), ti ∈ N0, AMs<sup>1</sup> and AMs<sup>2</sup> be the sets of all the allowed static markings of PNTS<sup>1</sup>

Symbolic representation of PNTS [PNTS1, PNTS2].COMP(t, ti) can be seen in Figure 2.


Symbolic representation of PNTS [PNTS1, PNTS2, …, PNTSn].SYNC(pi, po, ti, to, af1, …, afn, ti1, …, tin, tio) can be seen in Figure 3.

Lemma 3. Let PNTS<sup>1</sup> := (P1, T1, A1, AF1, TP1, TI1, IP1, OP1, RP1), PNTS<sup>2</sup> := (P2, T2, A2, AF2, TP2, TI2, IP2, OP2, RP2), …, PNTSn := (Pn, Tn, An, AFn, TPn, TIn, IPn, OPn, RPn) be arbitrary PNTSs, ∀i, 1 ≤ i ≤ n, ∀j, 1 ≤ j ≤ n: i 6¼ j ) PNTSi ∠ PNTSj, where n ∈ N, pi and po be arbitrary places, (pi ∉ P<sup>1</sup> ∪ P<sup>2</sup> ∪ … ∪ Pn) ∧ (po ∉ P<sup>1</sup> ∪ P<sup>2</sup> ∪ … ∪ Pn) ∧ (pi 6¼ po), ti and to be arbitrary transitions, (ti ∉ T<sup>1</sup> ∪ T<sup>2</sup> ∪ … ∪ Tn) ∧ (to ∉ T<sup>1</sup> ∪ T<sup>2</sup> ∪ … ∪ Tn) ∧ (ti 6¼ to), af1 ∈ N, af2 ∈ N, …, afn ∈ N, ti1 ∈ N0, ti2 ∈ N0, …, tin ∈ N0, tio ∈ N<sup>0</sup> and AMs1, AMs2, …, AMs<sup>n</sup> be the sets of all the allowed static markings of PNTS1, PNTS2, …, PNTSn. Let PNTS := [PNTS1, PNTS2, …, PNTSn].SYNC(pi, po, ti, to, af1, …, afn, ti1, …, tin, tio).

A critical path is then a designation for a sequence of activities whose time duration directly affects the time duration of the entire project. The activities that make up the critical path are then referred to as critical activities. There may be several critical paths in the project. When managing the project, a sequence of activities within a given network chart describing this project that increases the longest total time duration of a project is called its critical path. The critical path within the network chart can be used to determine the shortest time required to complete the project. The application of the CPM method can therefore determine which activities within the studied project are "critical" (i.e., activities on the longest path in the network chart describing the project) and which activities may be delayed in the execution of

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The special class CPNET ⊂ PNTS of PNTS is introduced in the following paragraphs to represent network chart used in the CPM method through PNTS. Special unary operator JOIN

Definition 10. The function JOIN: PNTS ! PNTS of net transition joining is defined as follows: if PNTS<sup>1</sup> := (P1, T1, A1, AF1, TP1, TI1, IP1, OP1, RP1) be the arbitrary PNTS, p ∉ P<sup>1</sup> be the arbitrary place, t1 and t2 be the arbitrary transitions, (t1 6¼ t2) ∧ (t1 ∈ T1) ∧ (t2 ∈ T1), ti ∈ N0, then PNTS := PNTS1.JOIN(p, t1, t2, ti), where PNTS PNTS := (P, T, A, AF, TP, TI, IP, OP, RP)

ix. RP := RP1. □ Symbolic representation of the unary operator JOIN application over the PNTS PNTS<sup>1</sup> can be

Definition 11. Let PNTS<sup>1</sup> := (P1, T1, A1, AF1, TP1, TI1, IP1, OP1, RP1), PNTS<sup>2</sup> := (P2, T2, A2, AF2, TP2, TI2, IP2, OP2, RP2), …, PNTSn := (Pn, Tn, An, AFn, TPn, TIn, IPn, OPn, RPn), where n ∈ N, be the arbitrary PNTSs, ∀i, 1 ≤ i ≤ n, ∀j, 1 ≤ j ≤ n: i 6¼ j ) PTSNi ∠ PTSNj. The class CPNET ⊂ PNTS

i. if p be an arbitrary place, p ∉ (P<sup>1</sup> ∪ P<sup>2</sup> ∪ … ∪ Pn), then PNTS BASEp ∈ CPNET, where

that is required in the definition of the class CPNET is introduced first.

the project without increasing its total time.

fulfills the following:

iii. A := A<sup>1</sup> ∪ {(t1, p), (p, t2)},

v. TP := TP<sup>1</sup> ∪ {(t, 1)},

vi. TI := TI<sup>1</sup> ∪ {(t1, p), ti)},

iv. AF := AF<sup>1</sup> ∪ {((t1, p), 1), ((p, t2), 1)},

then contains the following PNTSs:

BASEp := ({p}, ∅, ∅, ∅, ∅, ∅, p, p, ∅},

i. P := P<sup>1</sup> ∪ {p},

ii. T := T1,

vii. IP := IP1, viii. OP := OP1,

seen in Figure 4.

Figure 3. Symbolic representation of PNTS [PNTS1, PNTS2, …, PNTSn].SYNC(pi, po, ti, to, af1, …, afn, ti1, …, tin, tio).

If PNTS1, PNTS2, …, PNTSn are proper-formed, resp. well-formed, resp. pure-formed, PNTS and AMs = AMs<sup>1</sup> ⊗ AMs<sup>2</sup> ⊗ … ⊗ AMs<sup>n</sup> is the set of all the allowed static markings of PNTS PNTS, then also PNTS is proper-formed, resp. well-formed, resp. pure-formed, PNTS.

Proof. Clear, it directly follows from Definition 5, Definition 7 and Definition 9. □
