The Decision Problem for Some Finite Extensions of the Intuitionistic Theory of Abelian Groups

This pa, per continues the investigM, ions of [I] to [4] concerning the decision problem for v~u'ious ~flgebra, ie theories, formula, ted in Keyt.ing's predicate ea.leulus. Let T be the theory of M)elia, n groups with decidable equMity formula, ted in ]-[eytino"s predicate eMeulus HPC; theft, is, the la, n/ua, ge of T ha, s O, '- ,-a,nd the a, xioms of T ~n'e: (n) x+y-.q-~-x (b) (x+y)+z = .,'+(.q+:) (c) x+0 = .v (d) VxSy(x-F y-O) (g) x =y~y =x (h) (x =y)^(y =z)-~x =z (k) x =y-+(A(x)-~A(y)), A troy wff. In [1] it wa, s shown tlm, t the Theory T is ,ut~deeidable. It is of interest to ex,~mine the deeision problem for va, rious finite extensions of T. ~,re prove the following: T][]cO]~E~I 1. Let T,, be the extension1, of T with ~he axiom Vx(mx = 0). TheJl T,, is decidable ~ff m is the product of pairwise d'~ffere,~t prime n umbers. It is of interest to (.onsider the ~bo~,-e theories. In cl,~ssicM mathematics, we c,~n spea& of the first order theories of groups, for the groups ,%re given, and we ('~m stud) their first order properties. Intuitionistic mnthematics is not a,s developed yet :rod ill runny areas there is no a, greement :~s to wll{~t "the intuitionistic theory" is supposed to be. There is not a,s large ~ body of intaitionisticnlly constructed mn-themntics from which we ca, n a.bstrnet first order theories. Perhaps for this very l'e~tson, the role of ~xionm,tic theories in intuitionism is more prominent, and many bra.nches of intuitionistic mathem,%tics ~re first introduced a.s axiom systems 1)a, sed on (2nd) or ]st order ]Keyting's predicate ea.lculus. In the (.:~se of n, belinn groups, Heyting [9] formul~ted several algebraic