Mathematical Analysis of the Concept, equality.

Equality is one of the most important relations in mathematics and logic. That is why mathematicians and logicians have investigated properties of equality trying to achieve more clarity and precision in its understanding. In spite of their efforts, equality puzzle persists without consensus on any solution. To solve the problem of finding an adequate definition of equality, we suggest three innovations - the identification hierarchy, multifaceted scheme and relativistic approach. Developing this methodology, we explore the concept equality utilizing the hierarchy of identification relations used in mathematics and other fields. It is demonstrated that equality is a relative relation, which depends on the used perspective and occupies the middle position in the hierarchy of identification relations. Following this approach, we demonstrate how it is possible to introduce and discern different types and forms of equality the most basic of which are predicative, semiotic and systemic equalities. Various properties of equality are obtained.