We introduce the notions of weakly and locally positive Robinson theories. We give a characterization of weakly positive Robinson theories by the amalgamation property, and a syntactic characterization of locally positive Robinson theories.

Necessary and sufficient conditions are presented for the (first-order) theory of a universal class of algebraic structures (algebras) to have a model completion, extending a characterization provided by Wheeler. For varieties of algebras that have equationally definable principal congruences and the compact intersection property, these conditions yield a more elegant characterization obtained (in a slightly more restricted setting) by Ghilardi and Zawadowski. Moreover, it is shown that under certain further assumptions on congruence lattices, the existence of a model completion implies (...) that the variety has equationally definable principal congruences. This result is then used to provide necessary and sufficient conditions for the existence of a model completion for theories of Hamiltonian varieties of pointed residuated lattices, a broad family of varieties that includes lattice-ordered abelian groups and MV-algebras. Notably, if the theory of a Hamiltonian variety of pointed residuated lattices has a model completion, it must have equationally definable principal congruences. In particular, the theories of lattice-ordered abelian groups and MV-algebras do not have a model completion, as first proved by Glass and Pierce, and Lacava, respectively. Finally, it is shown that certain varieties of pointed residuated lattices generated by their linearly ordered members, including lattice-ordered abelian groups and MV-algebras, can be extended with a binary operation to obtain theories that do have a model completion. (shrink)

We study the relations between abelian groups B and C that every universal (resp. universal-existential) sentence true in B is also true in C, and give algebraic criteria for these relations to hold. As a consequence we characterize the inductive complete theories of abelian groups and prove that they are exactly the model-complete theories.

The underlying modules of existentially closed ▵-algebras are studied. Among other things, it is proved that they are all elementarily equivalent, and that all of them are existentially closed as modules if and only if ▵ is regular. It is also proved that every saturated module in the appropriate elementary equivalence class underlies an e.c. ▵-algebra. Applications to some problems in module theory are given. A number of open questions are mentioned.

A Boolean product construction is used to give examples of existentially closed algebras in the universal Horn class ISP generated by a universal classKof finitely subdirectly irreducible algebras such that Γa has the Fraser-Horn property. If ⟦a≠b⟧ ∩ ⟦c≠d⟧ = ∅ is definable inKandKhas a model companion ofK-simple algebras, then it is shown that ISP has a model companion. Conversely, a sufficient condition is given for ISP to have no model companion.

Minimality ranks in the style of Deissler are one way of measuring the structural complexity of minimal extensions of first-order structures. In particular, positive Deissler rank measures the complexity of the injective envelope of a module as an extension of that module. In this paper we solve a problem of the second author by showing that certain injective envelopes have the maximum possible positive Deissler rank complexity. The proof shows that this complexity naturally reflects the internal structure of the injective (...) extension in the form of the levels of the Matlis hierarchy. (shrink)

The model companion of the theory of fields with valuation and automorphism exists. A counterexample shows that the theory of models of ACFA equipped with valuation is not this model companion.

We prove the elimination of Magidor-Malitz quantifiers for R-modules relative to certain Q 2 α -core sentences and positive primitive formulas. For complete extensions of the elementary theory of R-modules it follows that all Ramsey quantifiers (ℵ 0 -interpretation) are eliminable. By a result of Baldwin and Kueker [1] this implies that there is no R-module having the finite cover property.

It is shown that the first-order theory Th R (A) of a countable module over an arbitrary countable ring R is ℵ 0 -categorical if and only if $A \cong \bigoplus_{t finite, n ∈ ω, κ i ≤ ω. Furthermore, Th R (A) is ℵ 0 -categorical for all R-modules A if and only if R is finite and there exist only finitely many isomorphism classes of indecomposable R-modules.