Combinations of structures

S. V. Sudoplatov

The Bulletin of Irkutsk State University. Series Mathematics, Tome 24 (2018), pp. 82-101

Résumé

We investigate combinations of structures by families of structures relative to families of unary predicates and equivalence relations. Conditions preserving $\omega$-categoricity and Ehrenfeuchtness under these combinations are characterized. The notions of $e$-spectra are introduced and possibilities for $e$-spectra are described. It is shown that $\omega$-categoricity for disjoint $P$-combinations means that there are finitely many indexes for new unary predicates and each structure in new unary predicate is either finite or $\omega$-categorical. Similarly, the theory of $E$-combination is $\omega$-categorical if and only if each given structure is either finite or $\omega$-categorical and the set of indexes is either finite, or it is infinite and $E_i$-classes do not approximate infinitely many $n$-types for $n\in\omega$. The theory of disjoint $P$-combination is Ehrenfeucht if and only if the set of indexes is finite, each given structure is either finite, or $\omega$-categorical, or Ehrenfeucht, and some given structure is Ehrenfeucht. Variations of structures related to combinations and $E$-representability are considered. We introduce $e$-spectra for $P$-combinations and $E$-combinations, and show that these $e$-spectra can have arbitrary cardinalities. The property of Ehrenfeuchtness for $E$-combinations is characterized in terms of $e$-spectra.

Détail

Keywords: combination of structures, $P$-combination, $E$-combination, $e$-spectrum.