Aspects of endomorphism monoids of certain algebras
Bibliographic record
Abstract
This thesis is concerned with the study of endomorphism monoids of certain algebras. We first describe the semigroup structure of a family of subsemigroups of the endomorphism monoid of an independence algebra A. Each of these subsemigroups is associated with a subalgebra B of A and is called the subsemigroup of endomorphisms with restricted range in B. Denoted by T(A,B), it consists of all endomorphisms of A whose image lies in B. We show in particular that such semigroups are not regular in general and that they present significant differences in their structure from that of End(A). \n \nIn a similar fashion, we investigate the semigroup structure of End(T_n), the endomorphism monoid of the full transformation monoid of a finite set with n elements. \nWe describe the ideals of End(T_n) and show that, in particular, T_n and End(T_n) are not respectively embeddable into each other (except in the degenerate case of n=1). \n \nWe then move on to the study of translational hulls of ideals of the endomorphism monoid of algebras. We start with the case of an independence algebra A, where we discuss the translational hull Ω(I) of the (0-)minimal ideal I of End(A). \nWe give conditions under which Ω(I) and End(A) are isomorphic and we construct a canonical isomorphism where possible. A more general approach of translational hulls in the case where A is an arbitrary algebra is then presented, where we prove that any ideal I of End(A) satisfying some representability and separability conditions on A will be such that its translational hull is isomorphic to End(A). \nFinally, we close this thesis by computing the translational hulls of some of the ideals of End(A), where A will stand either for a free algebra, an independence algebra, or T_n.
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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 | 0.001 |
| 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 | 0.003 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.000 | 0.002 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 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".