In one textbook appears the following sentence:
An algebraic structure is a nonempty set $M$ together with one or more operations (i.e. a function $*:M\times M\rightarrow M$) which satisfy some axioms. In other words, the definition of an algebraic structure is an axiomatic system. It can be proved that an axiomatic system which defines an algebraic structure verifies the logic requirements which any axiomatic system has to fulfill.
My question is: what does an algebraic structure need to fulfill (I mean the logic requirements)? It's about consistency or something like that? (The authors of the book say that the frame of the book doesn't allow that)
All I can do is give you some pointers to the standard terminology of elementary model theory and universal algebra: a structure consists of a set equipped with a set of finitary operations and relations (https://en.wikipedia.org/wiki/Structure_(mathematical_logic). An algebraic structure is one in which the only relation is equality (which many authors treat as a logical rather than mathematical concept) (https://en.wikipedia.org/wiki/Algebraic_structure). It is usual to associate a fixed set names for the operations and relations used in a structure. Such an association is called a signature for the structure and results in a logical language with function and relation symbols for the operations and relations. An axiomatic system is just a set of axioms in a logical language (https://en.wikipedia.org/wiki/Axiomatic_system). Any set of axioms in the language defined by some signature defines a class of structures for its signature (namely the structures in which those axioms hold) and conversely a class of structures for a signature defines a set of axioms (namely the axioms which hold in each structure in the class). A class of algebraic structures defined by a set of equational axioms is called an algebraic variety (https://en.wikipedia.org/wiki/Algebraic_variety).
There are no special requirements on axiomatic systems in normal usage, so in the last sentence in your quotation there is nothing to prove given the standard interpretations of the terminology involved.