Let \(A\) be a subset of a model \({\mathfrak M}\) of a first-order theory, and \(p(\overline x)\in S({\mathfrak M})\). The fundamental class \(\text{Cl}_A(p)\) over \(A\) is the set of all formulas \(\varphi (\overline{x}, \overline{y})\) with parameters from \(A\) such that there is \(\overline{m}\in{\mathfrak M}\) with \(\varphi (\overline{x},\overline{m})\in p\). If \(p\in S(A)\) and \(q\) is a type over a model extending \(p\), then \(q\) is an \(A\)-bound of \(p\) if \(\text{Cl}_A(q)\) is minimal possible. In a stable theory, all \(A\)-bounds for a given type \(p\in S(A)\) have the same fundamental class over \(A\), also called the \(A\)-bound \(\beta_A(p)\), and any \(q\) extending \(p\) does not fork over \(A\) if and only if \(\beta_A(q)= \beta_A(p)\).
If \({\mathfrak M}\subseteq A\), then \(p\in S(A)\) is a coheir of its restriction to \({\mathfrak M}\) if any \(\varphi (\overline{x})\in p\) is realized in \({\mathfrak M}\); it is an heir if \(\text{tp} (A/{\mathfrak M},a)\) is a coheir of \(\text{tp} (A/{\mathfrak M})\) for some (any) realization \(a\models p\). In a stable theory, any type over a model has a unique non-forking extension to any superset of the model; it is simultaneously an heir and a coheir.
Recently the machinery of forking and independence has been extended by B. Kim to the class of simple theories (which properly contains the class of stable ones). However, the above equivalences of non-forking no longer hold in this context. In their short and elementary paper, Lascar and Pillay characterize independence in a simple theory in the following way: if \(T\) is simple, then the following are equivalent:
1. \(a\) and \(b\) are independent over \(A\);
2. there is a model \({\mathfrak M}\) containing \(A\) such that \(\text{tp} (a/{\mathfrak M})\) is an \(A\)-bound of \(\text{tp} (a/A)\) and \(\text{tp} (a/{\mathfrak M},b)\) is a coheir of \(\text{tp} (a/{\mathfrak M})\);
3. any \(A\)-bound of \(\text{tp} (a/A)\) has the same fundamental class over \(A\) as \(\text{tp} (a/{\mathfrak M})\) for some model \({\mathfrak M}\) such that \(\text{tp} (a/{\mathfrak M},b)\) is a coheir of \(\text{tp} (a/{\mathfrak M})\).
Moreover, there is a unique maximal fundamental class over \(A\) for the nonforking extensions of a type \(p\in S(A)\).