On the Universal Theory of Classes of Finite Models1 By

First order theories for which the truth of a universal sentence on their finite models implies the truth on all models are investigated. It is proved that an equational theory has such a property if and only if every finitely presented model is residually finite. The most common classes of algebraic structures are discussed. 0. Introduction. John T. Baldwin in a review of the book Selected papers of Abraham Robinson. Volume 1, posed the following problem: "For what first order theories T does the truth of a universal sentence o on the finite models of T imply that o is consequence of TV Throughout this paper we will call such theories universally-finite. Baldwin's problem is suggested by Robinson's paper [19] where it is proved that the theory of Abelian groups and the theory of fields are universally-finite. See also Kueker [15]. We recall from well-known results [5] that a theory T is universally-finite iff every model of T can be embedded in an ultraproduct of finite models of T. However, such a characterization is difficult to handle even in simple cases. In this paper we look rather for a more useful characterization. First, we confine our attention to equational theories (§2). Then, we show that to every theory T in any language we can associate an equational theory E in a language without relation symbols in such a way that Baldwin's problem for T is reducible to Baldwin's problem of E (Proposition 3). The main theorem of this paper (see §2) proves that several statements are equivalent to the assertion that an equational theory T is universally-finite. The most important are: "Every finitely presented model of T is residually finite" and "Every quasi-identity true in all finite subdirectly irreducible models of T is true in all subdirectly irreducible models of T". We are convinced that our theorem provides a satisfactory characterization. In fact we get as an immediate consequence of it that the following varieties are universally-finite: Commutative unitary rings, commutative von Neumann regular rings with quasi-inverse as operation, lattices, fi-modules, where R is a finitely generated commutative unitary ring (this generalizes Robinson's result). On the other hand groups and unitary rings (cf. [6]) are examples of non-universally-linite varieties. Moreover, we characterize (§4) the varieties of i2-modules which are Received by the editors June 20, 1983. 1980 Mathematics Subject Classification. Primary 03C13, 03C05; Secondary 03C60, 08C10.