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This is a thesis that was defended in 2009 at Lomonosov Moscow StateUniversity.  In Chapter 1:  1. It is proved that that the class of lower (Skolem) elementary functions isthe set of all polynomial-bounded functions that can be obtained by acomposition of $x+1$, $xy$, $\max(x-y,0)$, $x\wedge y$, $\lfloor x/y \rfloor$,and one exponential function ($2^x$ or $x^y$) using formulas that have no morethan 2 floors with respect to an exponent (for example, $(x+y)^{xy+z}+1$ has 2floors, $2^{2^x}$ has 3 floors). Here $x\wedge y$ is a bitwise AND of $x$ and$y$.  2. It is proved that $\{x+y,\ \max(x-y,0),\ x\wedge y,\ \lfloor x/y \rfloor,\2^{\lfloor \log_2 x \rfloor^2}\}$ and $\{x+y,\ \max(x-y,0),\ x\wedge y,\\lfloor x/y \rfloor,\ x^{\lfloor \log_2 y \rfloor}\}$ are composition bases inthe functional version of the uniform $\mathrm{TC}^0$ (also known as$\mathrm{FOM}$).  3. The hierarchy of classes exhausting the class of elementary functions isdescribed in terms of compositions with restrictions on a number of floors in aformula.  The results of Chapter 1 are published in:  1) Volkov S.A. An exponential expansion of the Skolem-elementary functions,and bounded superpositions of simple arithmetic functions (in Russian),Mathematical Problems of Cybernetics, Moscow, Fizmatlit, 2007, vol. 16, pp.163-190  2) doi:10.1134/S1064562407040217  In Chapter 2 a simple composition basis in the class ${\cal E}^2$ ofGrzegorczyk hierarchy is described. This result is published in DOI:10.1515/156939206779238436  In Chapter 3 it is proved that the group of permutations$\mathrm{Gr}(Q)=\{f:\ f,f^{-1}\in Q\}$ is generated by two permutations formany classes $Q$. For example, this is proved for $Q=\mathrm{FP}$, where$\mathrm{FP}$ is the class of all polynomial-time computable functions (of thelength of input). The results of chapter 3 are published in DOI:10.1515/DMA.2008.046

Finite Bases with Respect to the Superposition in Classes of Elementary Recursive Functions, dissertation