Fixed - Point Logic. Quasi - Fractal Aspects

In this chapter we shall consider fixed - point logic, topological aspects, connected with fixed - point logic and quasi - fractal fixed - point logic. The main question is to fined examples shows that quasi - fractal fixed - point logic is useful in decision of practical complete tasks. It is shown that Shelah and Gurevich iterations, [1] are, in fact, contracting mappings in the fixed - point theorem. As the practical outcome we give in this Chapter some algebraic illustrations to [2]. For example, in the case of a fractal lattice, the limit of an approximate fractal lattice is a fractal lattice. The last sentence is illustrated by Theorem 4.12. We outline the sustainable subsystem of the system \(S\) which not distorted properties of the system \(S.\) In the Sect. 4.2.3 we introduce a notion of a quasi - fractal function. The practical outcome of a notion of a quasi - fractal function is the following: one can use this notion for modelling of simultaneous actions of a smart system. In Sect. 4.2.5 we get that the set of quasi - fractal algebraic lattices is everywhere dense in the quasi - fractal metric lattice space.