A note on weak dividing

We study the notion of weak dividing introduced by S. Shelah. In particular we prove that T is stable iff weak dividing is symmetric. In order to study simple theories Shelah originally defined weak dividing in [6] . This notion is overshadowed by that of dividing, as the first author proved that dividing is the right well-behaved notion for simple theories [2],[3],[5],and [4]. However Dolich’s paper[1] reminded us that weak dividing is still an interesting notion. There he noted that weak dividing is symmetric and transitive in stable theories, and that simplicity is characterized by the property that dividing implies weak dividing. Here we continue the investigation of the notion of weak dividing. Intriguingly, what weak dividing is to stability is analogous with what dividing is to simplicity. For example, we show that weak dividing is symmetric only in stable theories (2.5). Stability is also equivalent to the left local character of weak dividing. However for the transitivity of weak dividing, a similar analogy does not exist. Namely, in a non-stable simple theory (e.g. the theory of a random graph), weak dividing can be transitive (2.7). As usual, we work in a saturated model of an arbitrary complete theory T . Notation will be standard: a denotes a finite tuple, and M denotes a small elementary submodel. We assume that the reader has some familiarity with the basics of dividing/forking as in [3],[5] or [7].