Combinations of structures
We investigate combinations of structures by families of structures relative to families of unary predicates and equivalence relations. Conditions preserving ω-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 ω-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 ω-categorical. Similarly, the theory of E-combination is ω-categorical if and only if each given structure is either finite or ω-categorical and the set of indexes is either finite, or it is infinite and Ei-classes do not approximate infinitely many n-types for n ∈ ω. 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 ω-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.
About the Authors
Sergey V. Sudoplatov, Dr. Sci. (Phys.–Math.), Assoc. Prof., Leading Researcher, Sobolev Institute of Mathematics SB RAS, 4, Academician Koptyug Avenue, Novosibirsk, 630090, Russian Federation Head of Chair, Novosibirsk State Technical University, 20, K. Marx Avenue, Novosibirsk, 630073, Russian Federation Prof., Novosibirsk State University, 1, Pirogov st., Novosibirsk, 630090, Russian Federation, e-mail: firstname.lastname@example.org
03C30, 03C15, 03C50
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