On classes of structures axiomatizable by universal d-Horn sentences and universal positive disjunctions

摘要

We provide universal algebraic characterizations (in the sense of not involving any "logical notion") of some elementary classes of structures whose definitions involve universal d-Horn sentences and universally closed disjunctions of atomic formulas. These include, in particular, the classes of fields, of non-trivial rings, and of directed graphs without loops where every two elements are adjacent. The classical example of this kind of characterization result is the HSP theorem, but there are myriad other examples (e.g., the characterization of elementary classes using isomorphic images, ultraproducts and ultrapowers due to Keisler and Shelah).