Conditions for non-symmetric relations of semi-isolation

B. S. Baizhanov ; S. V. Sudoplatov ; V. V. Verbovskiy

Sibirskie èlektronnye matematičeskie izvestiâ, Tome 9 (2012), pp. 161-184

Résumé

We consider necessary and sufficient conditions for non-symmetric relations of semi-isolation in terms of colorings for neighborhoods of types, quasi-neighborhoods, and the existence of limit models. We show that, for any type $p$ in a small theory, its non-symmetry of isolation is equivalent to the non-symmetry of semi-isolation (where a realization $\bar a$ of $p$ isolates a realization $\bar b$ of $p$ and $\bar b$ does not semi-isolates $\bar a$) and is equivalent to the existence of a limit model over $p$. We generalize the Tsuboi theorem on the absence of Ehrenfeucht unions of pseudo-superstable theories and the Kim theorem on the absence of Ehrenfeucht supersimple theories for unions of pseudo-supersimple theories. We also present a survey of results related to non-symmetric semi-isolation.

Détail

Keywords: relation of semi-isolation, $(p,q)$-preserving formula, Ehrenfeucht theory, powerful type, quasi-neighborhood, coloring of a structure, strict order property, limit model.