On a conception of a mathematical system

In the first volume of the treatise “Éléments de mathématique” Bourbaki has given one of possible rigorous definitions of a mathematical structure. However, it turned out that this definition is not much suitable for broad usage, because structures in Bourbaki's sense possess considerable bracket partition, which means the definite order of priority in assignment of constituent partial structures (such as addition, order, norm, topology, measure, etc.). In this paper a conception of a mathematical system is presented, which includes as partial cases both the bracket conception of a mathematical structure in Bourbaki's sense and the bracket-free conception of an algebraic system in Birkhoff–Tarski's sense, and which enables in particular to construct corresponding bracket-free structures for many important mathematical systems.