Some operations on the family of equivalence relations

Abstract

Throughout the history of mathematics, the notion of an equivalence relation has played a fundamental role. It dates back at least to the time when the natural numbers first were introduced: a non-negative integer may be thought of as a representative of the equivalence class of sets with the same cardinality. To express such a simple and “obvious” fact with equivalence relations may seem unnecessarily cumbersome. Nothing is further from the truth. Equivalence relations play a decisive role as building elements in every area of mathematics. For instance, algebra is firmly founded on equivalence relations: groups theory, rings theory, modules and fields would basically be impossible to define and use without equivalence relations.