@MISC{Korczyński_mmlidentifier:e_siec., author = {Waldemar Korczyński}, title = {MML Identifier:E_SIEC.}, year = {} }

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Abstract

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. Let us mention that there exists a G-net structure which is EE. Let us note that there exists a G-net structure which is strict, GG, and EE. An E-net is an EE GG G-net structure. In the sequel N denotes an E-net. Next we state several propositions: (1) Let R, S be binary relations. Then 〈X,R,S 〉 is an E-net if and only if the following conditions are satisfied: