Characterization of Different Prime Bi-Ideals and Its Generalization of Semirings
Bibliographic record
Abstract
We introduce three sequences of different prime bi-ideals of semirings such that 11(12,13)-prime bi-ideal, 21(22)-prime bi-ideal and 31(32,33)-prime bi-ideal using bi-ideals. In this article, we characterize the different prime bi-ideals. We discuss that the 11-prime bi-ideal implies the 12-prime bi-ideal implies the 13-prime bi-ideal, but the reverse implication does not hold with the help of numerical examples. We investigate if a 21-prime bi-ideal implies a 22-prime bi-ideal, but the converse need not be true with the help of numerical examples. If G is any bi-ideal of a semiring S, then K(G) = {x ∈ G | x + y = z for some y, z ∈ G} is the unique largest k-bi-ideal contained in G. If Θ is a 21-prime bi-ideal of S, then Θ is a one-sided ideal of S. It is shown that there is a relation between G and K(G), in which G is a 13-prime bi-ideal. In our communication, 11-prime bi-ideal implies 21-prime bi-ideal. An interaction between a 31-prime bi-ideal implies a 32-prime bi-ideal, and a 32-prime bi-ideal implies a 33-prime bi-ideal; however, the reverse implication is invalid by some examples. Every 13-prime bi-ideal is a 22-prime bi-ideal, but the converse need not be true with the help of examples.
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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 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.005 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.000 | 0.002 |
| Insufficient payload (model declined to judge) | 0.004 | 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".