Kinds of Pregeometries of Cubic Theories
The description of the types of geometries is one of the main problems in the structural classification of algebraic systems. In addition to the well-known classical geometries, a deep study of the main types of pregeometries and geometries was carried out for classes of strongly minimal and $\omega...
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| Format: | Article |
| Language: | English |
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Irkutsk State University
2022-09-01
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| Series: | Известия Иркутского государственного университета: Серия "Математика" |
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| Online Access: | https://mathizv.isu.ru/en/article/file?id=1426 |
| Summary: | The description of the types of geometries is one of
the main problems in the structural classification of algebraic
systems. In addition to the well-known classical geometries, a
deep study of the main types of pregeometries and geometries was
carried out for classes of strongly minimal and $\omega$-stable
structures. These studies include, first of all, the works of
B.I.~Zilber and G.~Cherlin, L.~Harrington, A.~Lachlan in the
1980s. Early 1980s B.I.~Zilber formulated the well-known
conjecture that the pregeometries of strongly minimal theories are
exhausted by the cases of trivial, affine, and projective
pregeometries. This hypothesis was refuted by E.~Hrushovski, who
proposed an original construction of a strongly minimal structure
that is not locally modular and for which it is impossible to
interpret an infinite group. The study of types of pregeometries
continues to attract the attention of specialists in modern model
theory, including the description of the geometries of various
natural objects, in particular, Vamos matroids.
In this paper we consider pregeometries for cubic theories with
algebraic closure operator. And we notice that for pregeometries
$\langle S,{\rm acl}\rangle$ in cubic theories, the substitution
property holds if and only if the models of the theory do not
contain infinite cubes, in particular, when there are no finite
cubes of unlimited dimension. By virtue of this remark, we
introduce new concepts of $c$-dimension, $c$-pregeometry,
$c$-triviality, $c$-modularity, $c$-projectivity and $c$-locally
finiteness. And besides, we prove the dichotomy theorem for the
types of $c$-pregeometries. |
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| ISSN: | 1997-7670 2541-8785 |