Elementarily λ-homogeneous binary functions

Abstract

Let S and T be sets with S infinite, and ${*:S \times S \rightarrow T}$ a function. Further, suppose that λ is a cardinal such that ${\aleph_0 \leq \lambda \leq |S|}$. Say that (S, T, *) is elementarily λ-homogeneous provided (X, T,*) is elementarily equivalent to (Y, T, *) for all subsets X and Y of S of cardinality λ. In this note, we classify the elementarily λ-homogeneous structures (S, T, *). As corollaries, we characterize certain mathematical structures ${\mathfrak{S}}$ which are also “elementarily ${\lambda}$ -homogeneous” in the sense that all substructures of ${\mathfrak{S}}$ of cardinality λ are elementarily equivalent. Among our corollaries is a generalization of a theorem due to Manfred Droste.