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 machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.002 | 0.005 |
| Scholarly communication | 0.002 | 0.006 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.007 | 0.001 |
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 source (direct Gemma or distilled Codex), 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".