Symmetries structure waves in the wave language. The symmetries specify which modifications vary the set, and which do not. So symmetry defines a closed set. In the group theory sets are closed under some operations. For example, the inf mechanism for R (constructed in previous writings) specifies how the gate mechanism functions, and how finite numbers can diverge. So the inf mechanism specifies the closeness of the set of real numbers. Before, the inf mechanism was discussed as a collection of diagrams. Then summation or multiplication operations are defined based on those diagrams. In this writing, they are called diagram-based operations. Then the groups defined by them are diagram-based groups. Thus far there has been no difference between conventional groups and diagram-based groups. But this additional feature paves the way for further expansions.
The diagrams are constructed (at the lowest level) by OOP. Then on top of that diagram-based groups are constructed. So there can be several instances of the same diagram-based group based on the same collection of diagrams. Then those instances can interact with each other similar to waves. This expands the idea of isomorphism in groups. More accurately those instances are, in fact, different representations of the group. This means that a “vector space” (V space) should be defined for this purpose then vectors in that space are different elements of an instance of the group. Other instances of the same groups are also defined in the same V space and this enables interactions between elements in one instance of the group and also between different instances of that group (here each instance is considered as one entity and interacts with other entities – interactions in an instance of a group are similar to Homeostasis, and interactions between different entities can form neural networks ). Remember the main reason waves were used was to make such interactions possible.
Now the question is how to define and construct V space. A conventional vector space is constructed by base vectors. In diagram-based groups “based vectors” are basic modules and blocks in the OOP. So the objects at the lowest level construct the “basis vectors” for the V space.