Abstract
AbstractNecessary 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.
Highlights
The main aim of this paper is to understand what it means in algebraic terms for the theory of a universal class of algebraic structures to have a model completion
For classes that have finite presentations — including all quasivarieties, but not, for example, ordered abelian groups — a complete characterization was provided by Wheeler in [38] using the well-studied properties of amalgamation and coherence together with a more complicated property referred to as the conservative congruence extension property
The mentioned properties can be used to confirm that the theories of ordered abelian groups and linearly ordered MV-algebras have a model completion [36, 29], the conservative model extension property is, in general, rather difficult to prove or refute
Summary
Denote the class of linearly ordered members of V expanded with a binary operation ⊲ defined by x ⊲ y ≔ ye if e ≤ x otherwise. It follows from Jonsson’s Lemma [25] that V⊲ = ISP(Vc⊲). Let us define for any class K of algebras with a pointed residuated lattice reduct and a finite set of L-terms or L⊲-terms Γ ∪ {t},. K |= π → u ≈ v ⇐⇒ {s ≡ t | s ≈ t is an equation of π} |=K u ≡ v We use this notation to describe a deduction theorem for V⊲. Let V be any semilinear variety of pointed residuated lattices. For any finite set x and finite Γ ∪ {s, t} ⊆ TmL⊲(x),. Using the fact that V⊲ = ISP(Vc⊲), it suffices to prove that for any finite set x and finite Γ ∪ {s, t} ⊆ TmL⊲(x),.
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