Simplicity, and Stability in There

Firstly, in this paper, we prove that the equivalence of simplicity and the symmetry of forking. Secondly, we attempt to recover deenability part of stability theory to simplicity theory. In particular, using elimination of hyperimaginaries we prove that for any supersimple T , canonical base of an amalgamation class P is the union of names of-deenitions of P, ranging over stationary L-formulas in P. Also, we prove that the same is true with stable formulas for an 1-based theory having elimination of hyperimaginaries. For such a theory, the stable forking property holds, too. 1. Introduction The class of simple theories was introduced by Shelah 12]. After a period of neglect, the present author's Ph. D thesis and the joint paper with Pillay 8], motivated by Hrushovski's works on nite rank cases, spurred the rapid development of the study of simple theories. For the detailed background of simple theories, we refer to the survey paper appeared in the Bulletin of Symbolic Logic 7]. Accordingly, we assume that the reader has some familiarity with Shelah's stability theory, and simplicity theory appeared in 5],,8], and notions from 3]. But we state some of necessary deenitions below, and fundamental notions such as simplicity and forking will be recalled in section 2. During the last several years, one of the central problems in pure simplicity theory is whether the notions of Lascar strong type and strong type coincide. The more general question is whether any simple T has elimination of hyperimaginaries (see Deenition 1.3). This problem is related to the existence of canonical bases in M eq (see Deenition 1.1). For simple T, canonical bases exist as imaginaries if and only if T has elimination of hyperimaginaries 10]. Obviously in such T, elimination of hyperimaginaries implies the equivalence of the notions of Lascar strong type and strong type. B. Hart, A. Pillay and the present author showed the existence of canonical bases in the form of hyperimaginaries 3]. When they worked together at the Fields Institute in 1996, they also raised a question asking if forking is represented by stable formulas in simple T. These problems are still open, but there have been various important partial results. The present author proved that any small theory has elimination of hyperimaginaries (for nite tuples) 6]. Buechler and Shami independently proved that in any low theory, Lstp=stp 1]]11]. Recently Buechler, Pillay and Wagner 2] showed that any supersimple …