On the sheaf characterization of Gelfand rings
Bibliographic record
Abstract
This paper deals with the commutative rings with unit in which a+b = 1 implies that (1+ar)(1+bs) = 0 for suitable r and s. These rings have been considered, alternatively, in terms of conditions on their maximal ideals (DeMarco-Orsatti 1971, Mulvey 1979, Johnstone 1982) which are equivalent to the above in the presence of the Axiom of Choice (Banaschewski 2000). The purpose here is to give new characterizations of these rings as the rings of global elements of certain sheaves of rings on compact regular frames, the pointfree counterparts of compact Hausdorff spaces, augmenting a previous result of this type (Banaschewski 2000). In particular, one of the present cases provides the exact pointfree version of (the commutative case of) the original characerization of these rings of Mulvey (1979) which the earlier work did not achieve.
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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 | 0.002 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.003 | 0.001 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.002 | 0.006 |
| Open science | 0.000 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 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 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".