@MISC{PNPROC_1.MIZ,
  AUTHOR = {Bancerek, Grzegorz and Aoki, Mitsuru and Matsumoto, Akio and Shidama, Yasunari},
  TITLE = {Processes in {P}etri nets},
  SECTION1 = {Preliminaries},
  SECTION2 = {Markings on Petri Nets},
  SECTION3 = {Transitions and Firing},
  SECTION4 = {Firing Sequences of Transitions},
  SECTION5 = {Sequential Composition},
  SECTION6 = {Concurrent Composition},
  SECTION7 = {Neutral Process},
  DAY = {20},
  MONTH = {December},
  YEAR = {2002},
  ADDRESS1 = {Bia{\l}ystok Technical University},
  ADDRESS2 = {Shinshu University\\Nagano},
  ADDRESS3 = {Shinshu University\\Nagano},
  ADDRESS4 = {Shinshu University\\Nagano},
  SUMMARY = {Sequential and concurrent compositions of processes in Petri nets
are introduced. A process is formalized as a set of (possible), so called,
firing sequences.
In the definition of the sequential composition
the standard concatenation is used
 $$  R_1 \mathop{\rm before} R_2 =
 \{p_1\mathop{^\frown}p_2: p_1\in R_1\ \land\ p_2\in R_2\}  $$
The definition of the concurrent composition is more complicated
  $$
R_1 \mathop{\rm concur} R_2 =
 \{ q_1\cup q_2: q_1\ {\rm misses}\ q_2\ \land\
   \mathop{\rm Seq} q_1\in R_1\ \land\
   \mathop{\rm Seq} q_2\in R_2\}
  $$
For example,
  $$
 \{\langle t_0\rangle\} \mathop{\rm concur} \{\langle t_1,t_2\rangle\}
 = \{\langle t_0,t_1,t_2\rangle , \langle t_1,t_0,t_2\rangle , 
  \langle t_1,t_2,t_0\rangle\}
  $$
The basic properties of the compositions are shown.}}

