Extending types preserving diagram

The possibility of extending $D$-types while preserving the diagram is being studied. It is proved that if a diagram $D$ of some $\lambda$-homogeneous model and a family $S$ of $D$-types such that the domain of each of them has the cardinality less than $\lambda$ and is contained in a $D$-set $B$ are given, then there exists a generic extension $V$ of the original model $V_0$ of the set theory ZFC in which every type from $S$ extends to a $D$-type over $B$, and in the model $V$ all cardinals of the model $V_0$ not greater than $\lambda^+$ are preserved. Conditions under which all cardinals of the model $V_0$ are preserved in the model $V$ are specified.

УДК 510.67

Keywords: $D$-homogeneous model, $D$-type, extension of a $D$-type.