Under large cardinal assumptions, free LDAs with distinct finite generator counts are Σ₁-elementarily equivalent but not Σ₂-equivalent, and a homogeneous universal extension of the monogenerated free LDA is constructed where fixed applications are elementary embeddings.
Miller.Free Left Distributive Algebras
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Free Left Distributive Algebras and a Canonical Extension
Under large cardinal assumptions, free LDAs with distinct finite generator counts are Σ₁-elementarily equivalent but not Σ₂-equivalent, and a homogeneous universal extension of the monogenerated free LDA is constructed where fixed applications are elementary embeddings.