Geodesic
The lattice of definability. Origins and directions of research
Čebyševskij sbornik, Tome 22 (2021) no. 1, pp. 304-327

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The article presents results and open problems related to definability spaces (reducts) and sources of this field since the XIX century. Finiteness conditions and constraints are investigated, including the depth of quantifier alternation and the number of arguments. Results related to the description of lattices of definability spaces for numerical and other natural structures are described. Research methods include the study of automorphism groups of elementary extensions of the structures under consideration, application of the Svenonius theorem.
Keywords: definability, definability space, reducts, Svenonius theorem, quantifier elimination, decidability, automorphisms. 
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A. L. Semenov; S. F. Soprunov. The lattice of definability. Origins and directions of research. Čebyševskij sbornik, Tome 22 (2021) no. 1, pp. 304-327. http://geodesic.mathdoc.fr/item/CHEB_2021_22_1_a20/