Abstract:
For each $k$, $k\ge2$, three logical-functional languages are introduced
for the set of functions of $k$-valued logic: the positive expressibility
language $\operatorname{Pos}_k$,
the first-order language $1\operatorname{L}_k$,
and the second-order language $2\operatorname{L}_k$. On the basis of
the notion of expressibility in a language, the corresponding closure operators
are defined. It is proved that the operators of $1\operatorname{L}_k$-closure
and $2\operatorname{L}_k$-closure coincide. The $1\operatorname{L}_k$-closed and
$\operatorname{Pos}_k$-closed classes are described with the help of
symmetric groups and symmetric semigroups. The expressibility in the languages
$1\operatorname{L}_k$ and $\operatorname{Pos}_k$ is compared with
the parametric
expressibility and the expressibility by terms. The research was supported by the Russian Foundation for Basic Research,
grant 97–01–00989.
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