Definability of models by means of existential formulas without identity

Abstract

The problem of coding relations by means of a single binary relation is well known in the mathematical literature. It was considered in interpretation theory, and also in connection with investigations of decidability of elementary theories. Using various constructions (see, e.g., [2,6], proofs of Theorem 11 in [7] and Theorem 16.51 in [3]), for any model for a countable language, one can construct a model for ℒp (a language with a single binary relation symbol ) in which is interpretable. Each of the mentioned constructions has the same weak point: the universe of is different than the universe of . In [4] we have shown that, in the infinite case, one can eliminate this defect, i.e., for any infinite , we have constructed a model , having the same universe as , in which is elementarily definable. In all constructions mentioned above, it appears that formulas, which define in ( in ), are very complicated. In the present paper, another construction of a model for ℒp is given.