A New Construction Principle

2025/05/15 by Hyttinen, Tapani, Paolini, Gianluca, Quadrellaro, Davide Emilio
#03C48 #03C75 #06B25 #51E10 #51E12 #FOS: Mathematics #Logic (math.LO)

paper · doi:10.48550/arxiv.2505.10155

Abstract

We use the framework of Abstract Elementary Classes (AECs) to introduce a new Construction Principle CP(K,∗), which strictly generalises the Construction Principle of Eklof, Mekler and Shelah and allows for many novel applications beyond the setting of universal algebra. In particular, we show that CP(K,∗) holds in the classes of free products of cyclic groups of fixed order, direct sums of a fixed torsion-free abelian group of rank 1 which is not ℚ, (infinite) free (k,n)-Steiner systems, and (infinite) free generalised n-gons. From this we derive, in ZFC, that these classes of structures are not axiomatisable in the logic \mathfrakL_∞,ω1, and, under V=L, that they are not axiomatisable in \mathfrakL_∞,∞.

Citations

• A Module-theoretic Introduction to Abstract Elementary Classes
• On the Model Theory of Open Incidence Structures: The Rank 2 Case
• Deconstructible abstract elementary classes of modules and categoricity
• On the model theory of open generalized polygons
• Characterizing categoricity in several classes of modules
• Strongly Minimal Steiner Systems II: Coordinatization and Quasigroups
• First-Order Aspects of Coxeter Groups
• Some stable non-elementary classes of modules
• First-Order Model Theory of Free Projective Planes
• The Strong Small Index Property for Free Homogeneous Structures
• Beyond Abstract Elementary Classes: On The Model Theory of Geometric Lattices

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