## 4.5 Concatenated specific elementary processes

We have seen that a SEP (represented by a vector) is an even process, of defined variation of a state attribute, between two values - the initial and final state. If the final state of a process SEP1 (S02 in fig.4.5.1) becomes an initial state for other SEP2 process, we might say that the two processes are concatenated (or linked, in case of the temporal support processes, successive SEP). This abstract process of concatenation may go on as long as possible, specific and individual processes of any kind could be represented in this way, so in the same manner as any type of distribution, no matter how complex it is, can be decomposed into even elementary distributions.

Fig. 4.5.1

The figure 4.5.1 shows two such concatenated SEP, namely SEP1 and SEP2 which according to the relations 4.4.5 and 4.4.7 have a common component C12 = C(SEP1, SEP2) and the specific component (of SEP2 against SEP1) D12 = D(C12, SEP2) determined against the same reference direction of the previous SEP. The common component has the same direction with the reference direction and the specific component has the orthogonal direction oriented on the reference direction.

We may notice that two successive SEP shall have the same direction if there is no specific component (normal on the common direction) between the two SEP.

Comment 4.5.1: Introduction of the common and specific component concepts of two concurrent SEP and of their definition relations, provides new means for defining hard or incorrectly defined notions. One of these notions is for instance, the straight line definition. According to the elementary mathematics textbooks, the frequent definition of this abstract object is the following one: “The straight line is the shortest way between two points”. This definition mode enunciates an optimal property of the straight line (as a result of a selection between all the possible pathways between the two points) but not also the generation process of this kind of object. The utilization of the common and differential component notions of two SEP, which is also applicable for the concatenated processes, allows a processual definition of the straight line: “A series of concatenated first rank SEP, at which the specific component of two adjacent SEP is invariantly null is represented by a straight line”.

The cause for the occurrence of the differential components into a set of concatenated SEP shall be revealed in the following chapters or annexes. For the time being, we notice that the absence of such a component leads to an even process with the same direction, represented by a straight line according to the visual-graphical syntax, as we have aforementioned. This fact explains the graphical representation under the form of straight lines (axes) of the continuous set of values belonging to an one-dimensional attribute (this is obvious because since there is a single dimension, the differential component of the direction between two successive variations cannot exist), but this fact is also the basic explanation of the motion along a rectilinear trajectory of the material systems which are not subjected to the action of some forces with a normal component on its trajectory, that is a special situation - the absence of the action exerted by any force - corresponding to the motion of an isolated MS with an initial velocity.