Decision problem for separated distributive lattices

TLDR: It is proved here that the first-order theory of all separated distributive lattices is undecidable, and Rubin's result which made the undecidability proof very simple.

Abstract: It is well known that for all recursively enumerable sets X1, X2 there are disjoint recursively enumerable sets Y1, Y2 such that Y1 c X1, Y2 c X2 and Y1 U Y2 = X1 U X2. Alistair Lachlan called distributive lattices satisfying this property separated. He proved that the first-order theory of finite separated distributive lattices is decidable. We prove here that the first-order theory of all separated distributive lattices is undecidable. Introduction. A distributive lattice with 0 is separated if it satisfies the following separation property: for every x1, x2 there are Yi < x1 and Y2 < x2 such that Yil Y2 are disjoint (i.e. Yi A Y2 = 0) and Yi V Y2 x1 V x2. Alistair Lachlan introduced separated distributive lattices in [La] in connection with his study of the first-order theory of the lattice of recursively enumerable sets. He mentioned to me a question whether the first-order theory of separated distributive lattices is decidable. The answer is negative: in ?2 a known undecidable theory is interpreted in the firstorder theory of separated distributive lattices. The known undecidable theory is the first-order theory of the following structures: a Boolean algebra with a distinguished subalgebra. About undecidability of it see [Ru]. Actually the first version of the undecidability proof used the closure algebra CACD of Cantor Discontinuum, i.e. the Boolean algebra of subsets of Cantor Discontinuum with the closure operation. CACD is easily interpretable in the separated distributive lattice of functions f from Cantor Discontinuum into {0, 1, 2} such thatf1(2) is clopen. By [GS1] a finitely axiomatizable essentially undecidable arithmetic reduces to the first-order theory of CACD, hence to the first-order theory of the mentioned separated distributive lattice of functions, hence to the first-order theory of separated distributive lattices. The last step is somewhat complicated by the fact that [GS1] does not interpret the standard model N of arithmetic in CACD. (Even though [GS2] reduces the second-order theory of N to the first-order theory of CACD, [GS3] proves that N cannot be interpreted in CACD.) However the Boolean algebra of subsets of Cantor Discontinuum with a distinguished subalgebra of clopen (closed and open) sets is easily interpretable in CACD. This way I came to use Rubin's result which made the undecidability proof very simple. From the other side the cited result of [GS1] can be used to reprove Rubin's theorem and Received October 12 1980; revised August 30, 1981. 'The results were obtained and the paper was written during the Logic Year in the Institute for Advanced Studies of Hebrew University. ? 1983, Association for Symbolic Logic 0022-4812/83/4801-0020/$01.40