We show that if a first-order structure${\cal M}$, with universe ℤ, is an expansion of (ℤ,+,0) and a reduct of (ℤ,+,<,0), then${\cal M}$must be interdefinable with (ℤ,+,0) or (ℤ,+,<,0).

In 2001, the algebraico-tree-theoretic simplicity hierarchical structure of J. H. Conway’s ordered field${\mathbf {No}}$of surreal numbers was brought to the fore by the first author and employed to provide necessary and sufficient conditions for an ordered field (ordered$K$-vector space) to be isomorphic to an initial subfield ($K$-subspace) of${\mathbf {No}}$, i.e. a subfield ($K$-subspace) of${\mathbf {No}}$that is an initial subtree of${\mathbf {No}}$. In this sequel, analogous results are established forordered exponential fields, making use of a slight generalization of Schmeling’s conception of (...) atransseries field. It is further shown that a wide range of ordered exponential fields are isomorphic to initial exponential subfields of$({\mathbf {No}}, \exp )$. These include all models of$T({\mathbb R}_W, e^x)$, where${\mathbb R}_W$is the reals expanded by aconvergent Weierstrass system W. Of these, those we calltrigonometric-exponential fieldsare given particular attention. It is shown that the exponential functions on the initial trigonometric-exponential subfields of${\mathbf {No}}$, which includes${\mathbf {No}}$itself, extend tocanonicalexponential functions on theirsurcomplexcounterparts. The image of the canonical map of the ordered exponential field${\mathbb T}^{LE}$oflogarithmic-exponentialtransseries into${\mathbf {No}}$is shown to be initial, as are the ordered exponential fields${\mathbb R}((\omega ))^{EL}$and${\mathbb R}\langle \langle \omega \rangle \rangle $. (shrink)

An algebraically expandable (AE) class is a class of algebraic structures axiomatizable by sentences of the form $\forall \exists! \mathop{\boldsymbol {\bigwedge }}\limits p = q$. For a logic L algebraized by a quasivariety $\mathcal {Q}$ we show that the AE-subclasses of $\mathcal {Q}$ correspond to certain natural expansions of L, which we call algebraic expansions. These turn out to be a special case of the expansions by implicit connectives studied by X. Caicedo. We proceed to characterize all the AE-subclasses of (...) abelian $\ell $ -groups and perfect MV-algebras, thus fully describing the algebraic expansions of their associated logics. (shrink)

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)