A Decomposability Criterion for Elementary Theories

We prove that each elementary theory has a unique decomposition into indecomposable components and formulate a decomposability criterion. Definition 1 A theory T of signature Σ is called decomposable, if T is the deductive closure in the predicate calculus of signature Σ of all sentences of some theories S1 and S2 with the disjoint signatures Σ1 and Σ2, Σ1∪Σ2 = Σ (we use the notation: T = S1 ] S2). The theories S1 and S2 are called (decomposition) components of T . Only nontrivial decompositions, with Σ1 6= ∅ 6= Σ2, are of interest for consideration. Throughout this paper, we assume that every decomposition component of a theory T includes all equality formulas of T . Thus every component Si of signature Σi contains all sentences of T in signature Σi. For instance, if Σ consists of a sole symbol then every theory in this signature has only trivial decomposition. Let us formulate the main question under study: Consider a theory T of signature Σ defined by some set of axioms Φ in signature Σ. How can we determine whether T is decomposable judging from Φ? This question was formulated by D. Palchunov in [4]. The interest in this problem is connected with applications in computer science such as automated theorem proving [1] and the maintenance of terminological systems [3, 5]. ∗The author was supported by the RFBR (Grant 05–01–04003–NNIO a) and DFG project COMO, GZ: 436 RUS 113/829/0–1.