Definitions of Petri Net. Part II

The articles [3], [1], [4], and [2] provide the notation and terminology for this paper. In this paper x, y, X , Y are sets. We introduce G-net structures which are extensions of 1-sorted structure and are systems 〈 a carrier, an entrance, an escape 〉, where the carrier is a set and the entrance and the escape are binary relations. Let N be a 1-sorted structure. The functor echaos(N) yields a set and is defined by: (Def. 1) echaos(N) = (the carrier of N)∪{the carrier of N}. Let I1 be a G-net structure. We say that I1 is GG if and only if the conditions (Def. 2) are satisfied. (Def. 2)(i) The entrance of I1 ⊆ [: the carrier of I1, the carrier of I1 :], (ii) the escape of I1 ⊆ [: the carrier of I1, the carrier of I1 :], (iii) (the entrance of I1) · (the entrance of I1) = the entrance of I1, (iv) (the entrance of I1) · (the escape of I1) = the entrance of I1, (v) (the escape of I1) · (the escape of I1) = the escape of I1, and (vi) (the escape of I1) · (the entrance of I1) = the escape of I1. Let us observe that there exists a G-net structure which is GG. A G-net is a GG G-net structure. Let I1 be a G-net structure. We say that I1 is EE if and only if the conditions (Def. 3) are satisfied. (Def. 3)(i) (The entrance of I1) · ((the entrance of I1)\ idthe carrier of I1) = / 0, and (ii) (the escape of I1) · ((the escape of I1)\ idthe carrier of I1) = / 0.