Model Theory of Products of Chains

@misc{kurilic2026model,
  title = {{Model Theory of Products of Chains}},
  author = {Miloš S. Kurilić},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.24683},
  eprint = {2610.24683},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.24683},
}

TY  - UNPB
TI  - Model Theory of Products of Chains
AU  - Miloš S. Kurilić
PY  - 2026
DA  - 2026/10/17
PB  - arXiv
DO  - 10.48550/arxiv.2610.24683
UR  - https://arxiv.org/abs/2610.24683
AB  - We show that regarding the model-theoretic properties of distributive lattices isomorphic to finite products of linear orders there is a significant symmetry: if 𝕏_i, i<n are linear orders of size >1, 𝒯:=mathopTh (∏ _i<n𝕏 _i) and 𝒯_i:=mathopTh (𝕏_i), for i<n, then 𝒯 is finitely axiomatizable (resp. atomic, small) iff 𝒯_i, i<n, are finitely axiomatizable (resp. atomic, small); the countable atomic (resp. countably saturated) model of 𝒯, if it exists, is the product of countable atomic (resp. countably saturated) models of 𝒯_i; further, we describe all models of 𝒯: 𝕐 ⊨ 𝒯 iff 𝕐≅ ∏ _i<n𝕐_i, where 𝕐 _i ⊨ 𝒯_i, for i<n, and show that 𝔼 preccurlyeq ∏ _i<n𝕏_i iff 𝔼 = ∏ _i<n𝔼_i, where 𝔼_i preccurlyeq 𝕏_i, for i<n. (I(𝒯,ω)∈ { 1,c} was proved in a previous paper.) In addition, for each n∈ ℕ the class 𝒞_n of distributive lattices isomorphic to a direct product of n linear orders of size >1 is first order definable.
ER  -