Technological systems consist of interacting elements. There exist only two types of basic positive interactions between elements: either one element itself positively acts on another one or prevents/diminishes a harmful action on it.

Graphically, a positive action of element A on element B can be depicted as follows:

A ---> B,

whereas the situation where element A prevents/diminishes a negative/harmful action on element B can be depicted as follows:

---|---> B | | A

If A acts positively on B and this action entails a positive action of B on C, then we can speak about a positive action of A on C. This resembles the traditional order relation: from A < B and B < C it follows that A < C.

Analogously, if A prevents a harmful action on B and as a result B prevents a harmful action on C, then this also resembles an order relation but somewhat an opposite order relation to the previous one. That is why we denote it not by sign < but by sign >. In other words, A > B means A prevents a harmful action on B. Obviously, if A > B and B > C, then A > C.

However, our relations < and > are not the same as "less" and "greater" in mathematics. Here we cannot say that if A < B, then B > A. Then what can we say about the relation between A and C if A < B and C > B ?

If X < Y would imply that Y > X, then we would say that A < C. However, it is not the case here and in fact the following rules should be employed:

a) if A < B and C > B, then C > A. b) if A < B and B > C, then A > C.

Why these relations algebra is important ? There is a key concept in TRIZ called "the ultimate ideal solution". Roughly speaking, it says that everything should work on its own. In our case, if, say, A acts positively on B, then in the ideal situation B itself exerts this positive action on itself. In other words, in ideal system A and B should be merged or the functions of A should be delegated to B.

Up until now it was only clear what is "the ideal solution" for elements X and Y interacting directly (i.e. X ----> Y) or indirectly (i.e. X---->A---->B---->C----> .... ---->Y). The ideal solution for such elements was supposed to be their merger (as well as merger of all intermediate elements). However, it is less obvious what is "the ideal solution" for elements X and Y related, for example, as follows:

----|---->Y | | ---|---->C | | ---|-->B | | --|->A | | X

although one can conjecture that it also should be their merger as well as merger of all intermediate elements.

However, it is at all not obvious whether "the ideal solution" can be applied to elements X and Y related as follows:

X------>A<-------Y,

or as follows: ---|--->A<---|---- | | | | X Y

or as follows: X----->A<----|---- | | Y

Our algebra allows us to answer this question. Note that in the first diagram we had: X < A < B < C < .... < Y, and, hence, X < Y. The ideal solution for this diagram was merger of X and Y.

In the second diagram we had: X > A > B > C > Y, and, hence, X > Y. The ideal solution for this diagram was also merger of X and Y.

But in the third and fourth diagram we have X < A and Y < A, and X > A and Y > A respectively. Hence, in these cases we cannot say that either X < Y, or X > Y, or Y > X. On the other hand there are no examples of merger of such types of elements in the history of technology.

Our claim is that elements X and Y can be merged if and only if it can be shown that either X < Y, or X > Y, or Y > X.

Let us turn to the fifth diagram. Here we have X < A and Y > A. Hence, due to our laws of composition, we conclude that Y > X. Is there an evidence in the history of technology that elements X and Y from the fifth diagram eventually get amalgamated or one of them delegates its functions to another one ?

Yes, there is such an evidence. Consider, for example, the ancient transportation system consisting of horse, car, and wheels. The diagram of interactions between these elements looks as follows: pulls brakes HORSE------------->CAR<------|----------GROUND | |decrease |friction | WHEELS As we know, the functions of HORSE were eventually delegated to WHEELS with the advent of self-propelled cars.

By the way, if TRIZ were known 200 years ago, then one could easily predict (by resorting to the notion of "ultimate ideal solution") that cars will eventually move on their own without horses. However, this instrument would not allow one to predict that the functions of the mover will be delegated to wheels.

Our algebra would enable one to predict the both. Hence, it is a more powerful tool than "the ultimate ideal solution" and is in fact its generalization.