Some Theorems on the Lattice of Local Interpretability Types

In [4] J. MYCIELSKI introduced a very general notion of multidimensional local interpretability of first ordet theories. If we definethe relation ~ between theories T, S by T~ S iff Tis multidimensionaly locally interpretable in S, then ~is a. preordering. The induced partia.l ordering is a la.ttice ordering, it is called tke lattice of local interpretability iypes. We consider theories formalized in the first order logic without equality; equality is considered as a congruence and hence need not be interpreted absolutely. The languages of the theories are considered without function symbols, i.e. an n-ary function has to be included as a.n (n + l)-ary rela.tión. We regard two theories as equal iff they have the same theorems. A theoryT is locally interpretable in a theory S, in symbols T ~ S, iff each theorem of Tis interpretable in S. Equivalently: T ~ S uf each finite part of Tis interpretable in S. Theinterpretation may have parameters, variables may be translated as k-tuples of va.riables (then we speak about k-dimensional interpretations). By d~finition different theorems of T may have different interpretations of atomic relations in S. li thereis an interpretation of atomic relations common to all theorems of 'J', we speak a bout global interpretation of T fu S; insymbols T ~ 8 S. li each theor~m of T is interpre~a.ble k-dimensionally in S, we write T ~"S. The class Qf all theories T such that T ~ S and S ~ T forms the local interpretability type o/ S, denoted by ISI. 80 T E ISI iff T ~ S a.nd S ~ T. li we repla.ce in the previous paragraph ~ by ~8 (resp. by ~,,), we obta.in the definition of global (resp. k-dimensional) interpretability type o/ S. The relation ~ determines a partial ordering of the local interpretability types which is a distributive and complete lattice. The largest type is the type of the inconsistent theories, the lowest type is the type of the theories whose each theorem has a 1-point model. Among the types which do not contain incl;'nsistent theory, there is the largest type which is the type of theory Th(w, +, .) (as it is well known that for every consistent sentence we can define a model in arithmetic). There also exists the lowest type among the types which are not the lowest in the whole lattice. It is llie type of tneories having locally (i.e., each theorem has) finite models but which do not h~ve 1-point models. Hence it is, for example, the type of the theory based on the axiom (3x,y) (P(x) & -,P(y)). The sublattice of the lattice of the types which we obtain by tearing off the largest and the lowe;st types is therefore also a distributive and complete lattice. Let us (according to [1]) denote it by .,/t. More details can be found in [4], [o] and in the joint manuscript [1] (a revlsed version of it is being prepared for publication), where one can find also a number of first results and problems about the lattice .,/t. For completeness of the text some of these results are recapitulated in § 1.