{"id":"W2049256457","doi":"10.1007/s11083-012-9256-2","title":"A Hierarchy of Chains Embeddable into the Lexicographic Power $\\boldsymbol{({\\mathbb{R}}^\\omega,\\prec_{\\rm lex})}$","year":2012,"lang":"en","type":"article","venue":"Order","topic":"semigroups and automata theory","field":"Computer Science","cited_by":5,"is_retracted":false,"has_abstract":false,"ca_institutions":"York University","funders":"","keywords":"Lexicographical order; Combinatorics; Mathematics; Hierarchy; Omega; Tree (set theory); Order (exchange); Transfinite number; Sequence (biology); Branching (polymer chemistry); Chain (unit); Discrete mathematics; Binary tree; Physics","routes":{"ca_aff":true,"ca_fund":false,"ca_venue":false,"about_ca":false,"invisible_to_affiliation_only":false},"retraction":null,"screen":null,"direct_labels":[],"prediction":{"model_version":"codex-gemma-dda1882f352a","candidate_categories":[],"consensus_categories":[],"category_scores_codex":[0.0009700066,0.0002322908,0.0002674984,0.000188778,0.000213286,0.0001147699,0.001389998,0.0001100563,0.0001424495],"category_scores_gemma":[0.00007220374,0.0001560698,0.0001499695,0.001215802,0.0002212986,0.0008010779,0.000360346,0.0002352505,0.0001386252],"about_ca_system_candidate":false,"about_ca_system_consensus":false,"about_ca_system_score_codex":0.00002309226,"about_ca_system_score_gemma":0.00006786895,"about_ca_topic_candidate":false,"about_ca_topic_consensus":false,"about_ca_topic_score_codex":0.00008955581,"about_ca_topic_score_gemma":0.000008207468,"domain_scores_codex":[0.9981568,0.0001446633,0.0003216284,0.0003307249,0.0004142279,0.0006319431],"domain_scores_gemma":[0.9982178,0.0001973907,0.0001447336,0.001195613,0.00008733278,0.0001570942],"domain_codex":null,"domain_gemma":null,"domain_candidate":null,"domain_consensus":null,"study_design_codex":"theoretical_or_conceptual","study_design_gemma":"not_applicable","study_design_scores_codex":[0.00002192864,0.0006612376,0.005768611,0.00005997877,0.0001519995,0.000006783841,0.028483,0.00003212883,0.006676928,0.877507,0.007303854,0.07332656],"study_design_scores_gemma":[0.004545553,0.001099227,0.1191267,0.0005428544,0.0001735341,0.0002822153,0.005070451,0.0540286,0.03582866,0.1972248,0.5781327,0.003944732],"study_design_candidate":"theoretical_or_conceptual","study_design_consensus":null,"genre_codex":"empirical","genre_gemma":"empirical","genre_scores_codex":[0.7722297,0.00644842,0.1754568,0.00365054,0.001954273,0.0005671481,0.000006730276,0.0005555956,0.03913083],"genre_scores_gemma":[0.9893855,0.00005611066,0.0082307,0.0008632909,0.000123032,0.00003665626,0.000002142455,0.0000214462,0.001281065],"genre_candidate":"empirical","genre_consensus":"empirical","teacher_disagreement_score":0.6802822,"threshold_uncertainty_score":0.6364343,"prediction_status":"machine_predicted_unvalidated"},"machine_scores":{"provisional":true,"baseline":true,"maturity_gate_passed":false,"score_opus":0.008076845308624325,"score_gpt":0.2337108679808694,"score_spread":0.2256340226722451,"validation_status":"score_only:v0-immature-baseline","note":"Baseline scores from an immature model (maturity gate not passed). Scores rank; they never assert a category."}}