We consider a functioning property of a system with a finite set of elements and with different kinds of Boolean binary relations on it. We also construct the square matrices over arbitrary Boolean algebra which determine some Boolean binary relation and generate a cyclic semigroup with the maximum index and period. The looping of the system with a finite set of elements called an oscillator, is accompanied by appearing of subsequences (overtones) in a sequence of elements on the main diagonal of powers of a relevant Boolean matrix.
