In the paper all countable Boolean algebras with m distinguished ideals having countably-categorical elementary theory~ are described and constructed.
From the obtained characterization it follows that ~11 eountably-eategorical elementary theories of Boolean algebras with distinguished ideals are finite-axiomatizable, decidable and, consequently, their countable models are strongly constructivizable.
The subject of this paper is countable categoricity of elementary theories of Boolean algebras with distinguished ideals. These algebras are considered in the Signature cr = < 5~, w, --, O, 1, I1, ..., I,,> where, for l <~ m, I z is the unary predicate defining the ideal. The complete varia.nt of this paper is published in [7].
Earlier the following results were obtained: from Yu. L. Ershov's paper [2] it follows that the theory of Boolean algebra with distinguished ideal is decidable in, at least, the two following cases: 1) ~/I is finit% 2) there exists sup {x/I(x)} and ~ is atomic. M. Rabin in [8] proved that the theory of Boolean algebras with a distinguished ideal is decidable: A. S. 5Iorozov in [5] for any m, 0 < m < co constructed a Boolean algebra with the distinguished ideal the theory of which is nndecidable and the first characteristic equals m.
From the criterion of C. ]~yll-~ardzewski [9] it follows that the theory of a Boolean algebra 9~ is countably-categorical iff 9~ contains no more than a finite number of ntoms.
In the abstract [6] the author presents various examples of countably--categorical theories of atomless Boolean ~lgebras with a distinguished ideal and also the description of some classes of complete elementary theories O f Boolean algebras with the distinguished ideal is given there. It is also shown ~ that there exists a Continuum of various complete elementary theories oi an atomless Boolean algebra with the distinguished ideal even if the f~or by the distinguished ideal is:atomless.
In the abstract [4] P=-F. 5nrie alld A. Touraille showed that in ~ conntable Boolean algebra ?I one can choose a continuum of ideals with various elementary theories iff 9~ is not snperatomic. A similar result was obtained by the a~lthor.
In the present paper all countable Boolean algebras with distingui~
19. E. Pal'vhunov
& (x/r is an atom)
Acting in this way we construct a family of formulae V~ n ~ ~.
IJEzvi~ 4. The formulae V n are non-zero, non-vanishing, correct; they are either ideal or non-ideal; for ann i and j either Vi ~ V i or V i ~ V i ; if V~ and V i are obtained at different steps then V~ ~ V i iff V i < Vz for some 1 e Kim ~ where m is the number of the step at which V i is obtained.