ELEMENTARY EQUIVALENCE OF RIGHT-ANGLED COXETER GROUPS AND GRAPH PRODUCTS OF FINITE ABELIAN GROUPS
Bibliographic record
Abstract
Abstract. We show that graph products of finite abelian groups are elementarily equivalent if and only if they are ∃∀-equivalent if and only if they are isomorphic. In particular, two right-angled Coxeter groups are elementarily equivalent if and only if they are isomorphic. The notion of elementary equivalence is fundamental in model theory. One of the most natural problems about elementary equivalence, is given a class of algebraic systems, to understand which systems in this class are elementarily equivalent, i.e. to classify algebraic systems up to elementary properties. This problem in the case of groups is usually rather hard and there are only few examples known when this problem has a satisfactory solution. For the class of abelian groups, this is a well-known result of W. Szmielew (1955). For the class of ordered abelian groups this problem was studied by A. Robinson and E. Zakon (1960), M. Kargapolov (1963) and Yu. Gurevich (1964). A. Malcev (1961) solved this problem for classical linear groups. His approach was generalised by E. Bunina and A. Mikahlëv to other linear, algebraic and Chevalley groups. The problem of classifying linear groups over integers up to elementary properties was studied by V. Durnev (1995). For certain nilpotent groups this problem was studied by O. Belegradek, R. Deborah, A. Miasnikov, F. Oger, V. Remelsennikov. For certain free operator groups, this problem was solved by A. Miasnikov and V. Remeslennikov (1987). Around 1945 it was conjectured by A. Tarski, that elementary theories of free non-abelian groups of different rank coincide. This conjecture is now known as Tarski’s problem and has recently been solved by O. Kharlampovich and A. Miasnikov (2006), and, independently, by Z. Sela (2006). The classification of torsionfree hyperbolic groups up to elementary properties has been recently announced by Z. Sela. In this paper we address the classification of graph products of finite abelian groups up to elementary properties. We prove that two graph products of finite abelian groups are elementarily equivalent if and only if they are ∃∀-equivalent if
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How this classification was reachedexpand
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 | 0.001 | 0.001 |
| 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 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 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 teacher head, 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".