Remarks on model classes acting on one another

Abstract

An abstract notion of action of structures on one another was established by Marcev [2]: Let L and M be two first order languages having only the logical constants in common. Let R be a relational symbol not in LuM, and F a set of sentences in L u M u { R } (which is a two-sorted first order language). Then a s t ructure /~ for L is said to act on a structure 313 for M (in the sense of F) iff the disjoint union & o ] 3 constitutes a model of F under a suitable interpretation of R, where the range of the variables of L is g~_ and the range of the variables of M is Examples abound; we mention only a few: (i) L e t / k be a commutative ring with identity, ]3 an abelian group, and let R stand for unitary scalar multiplication, whose axioms constitute F. (ii) Let & be a group,]3 arbitrary, and let R stand for injective representation in the automorphism group of ]3 (cf. Rabin [3]). (iii) Let & be a lattice, ]3 arbitrary, and let F describe the embeddability of A into the lattice of all substructures (or congruence relations) of (iv) Let & be a field, ]3 an ordered abelian group, and let R stand for nontrivial valuation, whose axioms make up F. If 9.1 is a class of structures for L and ~5 a class of structures for M, let 9JF~B denote the class of all structures in 92 that act on at least one structure in ~ . Then we have the following theorem: