Model completions for universal classes of algebras: necessary and sufficient conditions

TL;DR

This paper characterizes when the first-order theory of universal algebra classes admits a model completion, linking algebraic properties with logical model-theoretic conditions, and applies these results to specific varieties like residuated lattices.

Contribution

It provides necessary and sufficient conditions for the existence of a model completion in universal algebra classes, extending previous characterizations and applying them to Hamiltonian varieties.

Findings

Conditions for model completion in algebraic theories are established.

Theories of lattice-ordered abelian groups and MV-algebras lack model completions.

Extensions of certain varieties can admit model completions with added operations.

Abstract

Necessary and sufficient conditions are presented for the (first-order) theory of a universal class of algebraic structures (algebras) to admit 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 admits 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 in order to obtain theories that do have a model completion.