Abstract
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).
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Acknowledgements
Open access funding provided by Johannes Kepler University Linz. We are grateful to an anonymous referee who provided helpful comments as well as the editor for a rather streamlined process. We also wish to thank Adam Přenosil for his suggestions on a previous version of this manuscript, and in particular for suggesting Example 3.4. An early incarnation of this paper was presented at the 94th Workshop on General Algebra (AAA94) in Novi Sad, Serbia, in June 2017. We are indebted to the audience there for their comments.
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This paper is dedicated to the loving memory of Carolina Blasio.
Guillermo Badia is supported by the project I 1923-N25 of the Austrian Science Fund (FWF), while João Marcos acknowledges partial support by CNPq and by the Humboldt Foundation.
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Badia, G., Marcos, J. On classes of structures axiomatizable by universal d-Horn sentences and universal positive disjunctions. Algebra Univers. 79, 41 (2018). https://doi.org/10.1007/s00012-018-0522-z
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DOI: https://doi.org/10.1007/s00012-018-0522-z