Different Partial Prime Bi-Ideals and Its Extension of Partial Ternary Semirings
Bibliographic record
Abstract
We discuss the partial bi-ideal of partial ternary semirings S. The partial bi-ideal is a new generalization of the ideal. To determine the relationships between the three types of partial prime bi-ideals and their examples. We constructed the partial right ideal, partial lateral ideal, partial left ideal, partial ideal, and partial bi-ideal generated by a single element. We interact with the relationships between HQ, LQ and RQ, where Q is bi-ideal. Consequently, we defined three distinct partial m-systems. The partial bi-ideal P of S is a partial 2-prime if and only if Z1Z2Z3⊆P, where Z1 is a partial right ideal, Z2 is a partial lateral ideal and Z3 is a partial left ideal of S implies either one of Z1⊆P, Z2⊆P and Z3⊆P. Also, we discuss HQ is a unique biggest two-sided partial prime ideal of S contained Q. Suppose that M is a partial m3-system and partial bi-ideal Q of S with Q∩M is empty, there exists a partial 3-prime P of S containing Q which includes P∩M=∅. Finally, examples were provided to illustrate the results.
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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 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 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".