Stable theories with a new predicate (Q2758050)

Let \(M\) be an \(L\)-structure and \(A\) be an infinite subset of \(M\). Then \(M^A\) denotes the structure on \(A\) which has a name \(R_\varphi\) for every \(\emptyset\)-definable relation \(\varphi(M) \upharpoonright A\). Its language is \(\{R_\varphi \mid\varphi\) is an \(L\)-formula\}. \(M_A\) denotes the enrichment of \(M\) with a new predicate for \(A\). \(A\) is small if \(N_B\equiv M_A\) for some \(B\) such that for every finite subset \(b\) of \(N\) every \(L\)-type over \(Bb\) is realized in \(N\). A formula \(\varphi(x,y)\) has the finite cover property (f.c.p.) in \(M\) if for all natural numbers \(k\) there is a set of \(\varphi\)-formulas \(\{\varphi (x,m_i)\mid i\in I\}\) that is not consistent in \(M\) such that every of its \(k\)-element subsets is consistent in \(M\). \(M\) has the f.c.p. if some formula has the f.c.p. in \(M\). The authors prove the following: Let \(A\) be a small subset of \(M\). Then, 1) If \(M\) does not have the f.c.p. then, for every \(\lambda\geq|L|\), if both \(M\) and \(M^A\) are \(\lambda\)-stable then \(M_A\) is \(\lambda\)-stable; 2) If \(M\) is stable and \(M^A\) does not have the f.c.p. then \(M_A\) is stable.