Three basic operations on labelled net structures are proposed: synchronised union, synchronised intersection and synchronised difference. The first of them is a version of known parallel composition with synchronised actions identically labelled. The operations work analogously to the ordinary union, intersection and difference on sets. It is shown that the universe of net structures with these operations is a distributive lattice and - if infinite pre/post sets of transitions are allowed - even a Boolean algebra. As a consequence, some representation theorems of this algebra are stated. The primitive objects are atomic net structures containing one transition with at most one pre-place or post-place (but not both). A simple example of a production system constructed by making use of the operations (and its transformations) is given. Some remarks on behavioural properties of compound nets are stated, in particular, how some constructing strategies may help to infer liveness. The latter issue is limited to semantics of place/transition nets without weights on arrows and with unbounded capacity of places and is not extensively investigated, since the main objective is focused on a calculus of net structures.

Keywords: net structures, synchronised operations, distributive lattice, boolean algebra, representation theorems

@article{ITA_2008__42_2_271_0, author = {Czaja, Ludwik}, title = {Calculi of net structures and sets are similar}, journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications}, pages = {271--284}, publisher = {EDP-Sciences}, volume = {42}, number = {2}, year = {2008}, doi = {10.1051/ita:2007033}, zbl = {1144.68042}, mrnumber = {2401262}, language = {en}, url = {http://www.numdam.org/articles/10.1051/ita:2007033/} }

TY - JOUR AU - Czaja, Ludwik TI - Calculi of net structures and sets are similar JO - RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications PY - 2008 DA - 2008/// SP - 271 EP - 284 VL - 42 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ita:2007033/ UR - https://zbmath.org/?q=an%3A1144.68042 UR - https://www.ams.org/mathscinet-getitem?mr=2401262 UR - https://doi.org/10.1051/ita:2007033 DO - 10.1051/ita:2007033 LA - en ID - ITA_2008__42_2_271_0 ER -

%0 Journal Article %A Czaja, Ludwik %T Calculi of net structures and sets are similar %J RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications %D 2008 %P 271-284 %V 42 %N 2 %I EDP-Sciences %U https://doi.org/10.1051/ita:2007033 %R 10.1051/ita:2007033 %G en %F ITA_2008__42_2_271_0

Czaja, Ludwik. Calculi of net structures and sets are similar. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Volume 42 (2008) no. 2, pp. 271-284. doi : 10.1051/ita:2007033. http://www.numdam.org/articles/10.1051/ita:2007033/

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