On elementarily κ -homogeneous unary structures
Greg Oman · Forum Mathematicum · 2010
Abstract Let L be a first-order language with equality and let 𝔘 be an L -structure of cardinality κ . If ℵ 0 ≤ λ ≤ κ , then we say that 𝔘 is elementarily λ -homogeneous iff any two substructures of cardinality λ are elementarily equivalent, and λ -homogeneous iff any two substructures of cardinality λ are isomorphic. In this note, we classify the elementarily λ -homogeneous structures ( A , ƒ) where ƒ : A → A is a function and λ is a cardinal such that ℵ 0 ≤ λ ≤ | A |. As a corollary, we obtain a complete description of the Jónsson algebras ( A , ƒ), where ƒ : A → A .