On the structure of a generalization of weakly associative lattices.
Bibliographic record
Abstract
The concept of weakly associative lattices (i.e. relational systems with a reflexive and antisymmetric relation ≤, in which for each pair of elements there exist a least upper and a greatest lower bound) was introduced in [3] and [5]. In [4] WU-systems are defined, i.e. weakly associative lattices with the unique bound property, and their equivalence with projective planes is described. In this paper we introduce WUλ -systems, and discuss their relation to symmetric 2−(v, k, λ) designs equipped with a special “loop-free” mapping. The concept of weakly associative lattices (i.e. relational systems with a reflexive and antisymmetric relation ≤, in which for each pair of elements there exist a least upper and a greatest lower bound) was introduced in [3] and [5]. Ervin Fried and Vera T. Sos defined WU-systems in [4]; i.e. weakly associative lattices with the unique bound property; their equivalence with projective planes is described in [4]. In this paper we introduce WUλ systems (here ‘U’ stands for ‘uniform’), and discuss their relation to symmetric 2−(v, k, λ) designs equipped with a special “loop-free” mapping. Definition. Let V = 〈V,≤〉 be a system with a reflexive and antisymmetric relation ≤. Let U(v) = {w ∈ V : v ≤ w} and L(v) = {w ∈ V : w ≤ v} for each v ∈ V . Given a positive integer λ, we call V a WUλ -system if for each v1 6= v2 ∈ V both of the sets U(v1) ∩ U(v2) and L(v1) ∩ L(v2) has size λ. Note that for λ = 1 we get WU-systems of [4]. Remark. The Paley tournaments for (prime power) q = 4k − 1 give a series of examples for WUλ -systems with λ = (q + 1)/4. (V = GF (q), and for a, b ∈ GF (q) define a ≤ b iff (b− a) is a square element of GF (q).) Definition. A 2 − (v, k, λ) design (sometimes they are denoted by Sλ(2, k, v)) is an incidence structure D = (P,B), where the elements of P are called points, the elements B ∈ B are subsets of P called blocks, and |P| = v, |B| = k for all B ∈ B, and for any pair of distinct points there are exactly λ blocks containing both of them. A design is called symmetric, if |P| = |B|; in this case v = k(k−1) λ + 1 holds ([2]). ∗Research was partially supported by OTKA and Eotvos grants.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".