Strongly Flat and<i>PO</i>-Flat<i>S</i>-Posets
Why this work is in the frame
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Bibliographic record
Abstract
ABSTRACT For a monoid S, a (left) S-act is a nonempty set B together with a mapping S × B → B sending (s, b) to sb such that s(tb) = (st)b and 1b = b for all s, t ∈ S and b ∈ B. Over the past three decades, an extensive theory of flatness properties has been developed (involving free acts, projective acts, strongly flat acts, Condition (P), flat acts, weakly flat acts, principally weakly flat acts, and torsion free acts). A recent and complete discussion of this area is contained in the monograph Monoids, Acts and Categories by Kilp et al. (2000 Kilp , M. , Knauer , U. , Mikhalev , A. V. ( 2000 ). Monoids, Acts and Categories, with Applications to Wreath Products and Graphs . Berlin , New York : Walter de Gruyter .[Crossref] , [Google Scholar]). Partially ordered acts over a partially ordered monoid S, or S-posets appear naturally in the study of mappings between posets. Preliminary work on flatness properties of S-poset, was done by Fakhruddin in the 1980s (see Fakhruddin, 1986 Fakhruddin , S. M. ( 1986 ). Absolute flatness and amalgams in pomonoids . Semigroup Forum . 33 : 15 – 22 . [CSA] [Crossref], [Web of Science ®] , [Google Scholar] 1988 Fakhruddin , S. M. ( 1988 ). On the category of S-posets . Acta Sci. Math. (Szeged) . 52 : 85 – 92 . [CSA] [Google Scholar]), and continued in recent (Bulman-Fleming and Laan, 2005 Bulman-Fleming , S. , Laan , V. ( 2005 ). Lazard's Theorem for S-posets . Math. Nachr. 278 : 1 – 13 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]; Shi et al., 2005 Shi , X. P. , Liu , Z. K. , Wang , F. G. , Bulman-Fleming , S. ( 2005 ). Indecomposable, projective and flat S-posets . Comm. Algebra. 33 : 235 – 251 . [CSA] [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). In Bulman-Fleming and Laan (2005 Bulman-Fleming , S. , Laan , V. ( 2005 ). Lazard's Theorem for S-posets . Math. Nachr. 278 : 1 – 13 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]), the Stenström-Govorov-Lazard theorem was shown in the context of S-posets. Tensor products of S-posets, free, projective and flat S-posets, as well as an analogue of Condition (P) and Condition (E) were introduced in these articles, but strongly flat, flat, weakly flat, principally weakly flat acts and torsion free S-posets were not considered. The present article addresses these matters.
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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 | 0.001 |
| Open science | 0.003 | 0.002 |
| 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 it