Degrees of presentability of structures. I

Presentations of structures in admissible sets, as well as different relations of effective reducibility between the structures, are treated. Semilattices of degrees of $\Sigma$-definability are the main object of investigation. It is shown that the semilattice of degrees of $\Sigma$-definability of countable structures agrees well with semilattices of $T$- and $e$-degrees of subsets of natural numbers. Also an attempt is made to study properties of the structures that are inherited under various effective reducibilities and explore how degrees of presentability depend on choices of different admissible sets as domains for presentations.