Can a structure be carrier-free? Philosophy says that it is impossible. However, contemporary theoretical physics gives a different answer.
In mathematics, there exist various concepts of a structure: the structure genus of a structure (a rather sophisticated definition of Bourbaki), a lattice (an algebraic notion generalizing a Boolean algebra), a topological structure, a geometric structure, etc. For physical applications, I would propose a mathematical definition of the structure as an n-ary relation on a set defined by some subset of an n-product of this set. This concept correlates with the definition of Bourbaki in some way and absorbs other definitions of a structure. In particular, morphisms of a set are structures in this sense. Nevertheless, in all existent variants, a mathematical structure is introduced on a carrier set.
In physics, however, it appears that a set, carrying a structure, often itself consists of elements of some structure. For example, a classical field, defined as a section of a fibre bundle, is a morphism. i. e., a structure, called the geometric structure. It is obvious, that quantum operators as elements of a certain algebra exemplify an algebraic structure. Moreover, by virtue of the well-known GNS construction in algebraic quantum theory, a Hilbert space of states, which quantum operators act in, consists of equivalence classes of these operators possessing the same average value, and so, it also is a set of elements of an algebraic structure.
Contrary, a point mass in classical mechanics is not part of any structure. However, in modern united models of fundamental interactions, a quantum field acquires a mass as a result of its interaction with a Higgs field. It follows that a mass is a derivative characteristics of two structures. Thus, the massive matter ceases to be a fundamental concept. For example, a particle and an antiparticle, annihilating, are converted into photons.
At present, with theoretical and mathematical viewpoint, all known fundamental physical objects are structures whose carrier consists of elements of some other structure having its carrier another structure, etc. Moreover, a structure can be defined on different carriers or be carrier-free. For instance, morphisms of some vector space are representations of a certain abstract group which is defined for itself and admits other representations.
If all physical objects, e.g., classical and quantum fields are a structure, then what is a carrier of a structure in the physical world? Is there such a carrier? Of course, the matter does not disappear, but is somewhat illusory.