تقرير
On the first-order theories of quaternions and octonions
| العنوان: | On the first-order theories of quaternions and octonions |
|---|---|
| المؤلفون: | Savi, Enrico |
| سنة النشر: | 2024 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Algebraic Geometry, Mathematics - Logic, Mathematics - Rings and Algebras, 03C10, 03C98 (Primary), 16K20, 17A35, 30G35, 14P10 (Secondary) |
| الوصف: | Let $L$ be the language of rings. We provide an axiomatization of the $L$-theories of quaternions and octonions and we characterize the models of mentioned theories: they coincide, up to isomorphism, to quaternion and octonion algebras over a real closed field, respectively. We prove these theories are complete, model complete and they do not have quantifier elimination. Then, we focus on the class of ordered polynomials. Over $\mathbb{H}$ and $\mathbb{O}$ these polynomials are of special interest in hypercomplex analysis since they are slice regular. We deduce some fundamental properties of the zero loci of ordered polynomials from completeness and we prove the failure of quantifier elimination for the fragment of ordered formulas. Comment: The introduction has been substantially enlarged, the structure of the paper slightly changed in simplifying some parts and clarifying others. The layout also changed, it is way larger, that's why the number of pages decreased even though the paper is richer. 19 pages, comments are very welcome |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2404.04976 |
| رقم الأكسشن: | edsarx.2404.04976 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |