O S PSEUDO-so-CATEGORIC~4L THEORIES by ~S A L I S A MARCJA and CARLO TOFFALORI in Firenze (Italy)') S 0. Introduction -4 countable, complete, quantifier eliminablr first order theory I' is said to be pseudo-%-cnteyorical ( x an infinite cardinal) if for every !JJl, % 1 T, (where B(!JJl). B((31) drnote the atomic Boolean algebras of definable subsets of M , M ; see 151). For x 2 sl . T is pseudo-x-categorical iff T is x-categorical (see [6]) ; obviously, the same ir not true for K = so : so-categorical theories are pseudoso-categorical, but not conversely. In t hi. pnper. 11 e approach the description of preudos,-categorical theories, showing in particular that : i) s 1 -categorical theories are p~~udo-s0-cateporica1 ;
ii) erery countable atomic Boolean algebra i q komorpliic to the Boolean algebra of definable subsets of a countable model of a countable superstable pseudo-so-categorical theory.
We assume familiarity with [5]. and with the usual concepts of theory of model. and Boolean algebras. The notation 11 ill be thr uhual one.
We wish to thank GREGORY CIIERLIN for his valuable contributions and sugge\tions.
81. Pseudo-N,-categoricity snd s -categoririty
We recall some basic facts concerning the classification of isomorphisni types of Boolean algebras. Let B be a countable Boolean algebra, we define a sequence of ideals
I,(B) of B ( A an ordinal) by induction on A :
a ) Io(B) = (0); b) if 1 = Y + 1 and I is the ideal of B/Z,(B) generated by the atoms of B / I v ( B ) . then c) if I. is a limit ordinal, set I,(B) = U ZJB). * < a An obvious argument shows that there exists an ordinal il < cul such that I , ( B ) = = I,+'(B), let p be the first ordinal with this property, then I,(B) = I,(B) for all A 2 p. If B is a superatomic Boolean algebra. then I,@) = B. p is a successor ordinal, and, if p = Y + 1, then B/I,(B) has finitely many atoms. Let us indicate in this case with cxB the greatest ordinal such that I J B ) + B (cxg < u l ) , and with d, the (finite) number of the atoms in B/Imn(B); aB, dB are called the CBrank, CB-degree of B (CB = Cantor-Bendixson); then the ordered pair (a,, d,) c o y -let Z,(B) be the set of preimages in B of elements of I; l ) Work performed under the auspices of Italian C.N.R. (G.N.S.A.G.A.