On a presentation of Petri nets and their morphisms

Working with system-designing one has often to transform a simple, but usually not readable model of a system to a more complicated but easier to analyzing form. In the paper Petri nets of a class are represented as members of an equationally deened class of structures. This representation leads to a relatively simple description of some typical constructions on Petri nets e.g. products and coproducts, and their transformations which play an important role as tools for (de)composition of systems. Perhaps the most often used presentation of graphs deenes them as algebras of the form G=(ver(G),arr(G),d 0 ,d 1) with ver(G) being the set of vertices of the graph G, arr(G) being the set of its arrows and incidence operations d 0 ,d 1 : arr(G)!ver(G).The presentation of graphs used in the paper deenes graphs with disjoint sets of vertices and arrows as equationally deenable algebras with one carrier-set only. By a graph we mean any triple G=(X(G),s,t) with a set X(G) (the carrier of G) and unary source and target operations s, t: X!X satisfying the conditions s(s(x))=t(s(x))=s(x) & s(t(x))=t(t(x))=t(x). The passing from this deenition to the two-sorted presentation mentioned above is via equations: ver(G)=fx2X: s(x)=t(x)=xg, arr(G)=Xnver(G), d 0 =snid ver(G) , d 1 =tnid ver(G). The class of homomorphisms of such graphs is richer than the class of "clas-sical" homomorphisms of graphs. If one wants to use graph homomorphisms as a model of an "aggregation of arrows into vertices" one has to choose between the simple two-sorted deenition of graphs and complicated deenition of their homomorphisms, or a "not typical" deenition of graphs (in this case of the above one-sorted presentation) and typical algebraic deenition of their homo-morphisms. If we do not want to distinguish vertices and arrows of a graph then some diierent sequences of elements of a graph may represent the same path in the \normal" meaning of this word. Such sequences of elements of a graph will be called routes. A route in a graph G=(X,s,t) is any nonempty sequence p=x 1 x 2 ...x n of elements of the carrier set of G such that t(x i?1)=s(x i) for every in. A route can be seen as a description of just a route; walking on a graph we go along the arrows and visit vertices. An occurrence of a vertex may be seen as a \unit" of such a visit. This is of course a …