SIMPLE PAIRS OF EQUIVALENCE RELATIONS by CARLO TOFFALORI in Camerino (Italy) 8 1. Introduction In f6l and (71 we considered theories of two equivalence relations Eo, E l such that if E dethere is an h E w such that for all u the E-class of u contains at most h classes 01 either Eo or E,. notes the equivalence relation generated by E, and E l , then ( + ) Notice that (+) implies that E is 0-definable, as E = R Z h . where RZI(v, w) means
V?E(O,IJ~~O...~ZZ~(ZO= 0 A Z Z k = W A A:<~(E~(ZZI> ZZ:+I)" E I -~( ~z I + I I ZZI+Z))).
On the other hand, E = RZh can be expressed by a suitable first order sentence in the language with two relations symbols, as it suffices to state R2(k+1
then also (+) can be written by a first order sentence.
We studied these theories from the point of view of the stability theory. Our main result was that they are always classifiable according to SHELAH (namely superstable, presentable, shallow and with the existence property, see [l]), and every type has finite U-rank, in fact every I-type has U-rank 6 3; moreover one can show that RC(v = v) s 3, where RC denotes the Shelah degree [I] (see the next section for a proof). However, among these theories there exist some non-w-stable examples, and in some sense this says that they are not "so simple", at least with respect to the Stability Spectrum Theorem (see Theorem 111.4. 36 in [l]). But lately a much finer notion of "simple" theory has been proposed within the "geometrical" stability theory. A clear introduction to this matter can be found in [3]. We recall here that if T is a theory satisfying RC(v = u ) < w (and this is our case), then it is reasonable to assume that T is locally simple iff any pregeometry (D, cl), where D is a minimal set, is locally modular, while T is globally simple iff T is weakly normal.
It may be worth reminding here the definition of these notions.
Let U denote the monster model of T. If X 5 U, p is a (possibly incomplete) type over X and D is the set of realizations of p , then D is said to be minimal if (D is infinite and) for all definable sets D' either D n D' or D -D' is finite. With no loss of generality, p is closed under finite conjunctions. Notice that D defines a (complete) type p D over X; p D contains exactly those formulas defining a set D' such that D -D is finite. Moreover, for every definable D', D n D' is finite if and only if there is d(u) E p such that d(u) A D' is finite. Define now a 26 Zeitschr. f. math. Lngik