My article “What is a
mathematical structure” (2013) came out

**(#)**.
"A notion of the mathematical
structure was introduced at the beginning of XX century. However, for a long
time, mathematical objects were believed to be given always together with some
structure, not necessarily unique, but at least natural (canonical). And only a
practice, e.g., of functional analysis has led to conclusion that a canonical
structure need not exist. For instance, there are different “natural”
topologies of a set of rational numbers, different smooth structures of a four-dimensional
topological Euclidean space, different measures on a real line, and so on.

In mathematics, different
types of structures are considered. These are an algebraic structure, a
topological structure, cells whose notion generalizes the Boolean algebras and
so on. In the first volume of their course, Bourbaki provide a description of a
mathematical structure which enables them to define “espece de structure” and,
thus, characterize and compare different structures. However, this is a
structure of mathematical theories formulated in terms of logic. We aim to suggest a wider
definition of a structure which absorbs the Bourbaki one and the others, but
can not characterize different types of structures. This definition is based on
a notion of the relation on a set, and it generalizes the definition of a
relational system in set theory.

Morphisms and functions are
structures in this sense that provides a wide circle of perspective applications of
this notion of the structure to mathematical physics.

In particular, let us mention
the notions of the universal structure on a set (see Section 2) and the abstract
structure on its own elements. One can show that any structure is a constituent
of a universal structure, and that any structure admits an exact representation
as a constituent of some abstract structure.

Though we follow the von Neumann – Bernays – Gödel set theory,
structures on sets only are considered unless otherwise stated. This is sufficient
in order to investigate real, e.g., physical systems."

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