秩为$3$的泛齐次单拟阵
A Universal Homogeneous Simple Matroid of Rank $3$
摘要：
本文构造了秩为$3$的$\wedge$ -齐次泛单拟阵，即一个可数单秩~ $3$拟阵$M_*$，其中$\wedge$ -嵌入每一个有限单秩$3$拟阵，并且使得$M_*$的有限$\wedge$ -子几何之间的每一个同构扩张为$M_*$的一个自同构.我们还构造了一个$\wedge$ -齐次拟阵$M_*(P)$，它对于有限单秩$3$拟阵类是通用的，它省略了一个给定的有限射影平面$P$ .然后证明了这些结构不是$\aleph_0$范畴的，它们具有独立性，它们具有平稳独立关系，并且它们的自同构群嵌入对称群$Sym(ω)$ .最后，利用$M_*$的自由射影扩张$F(M_*)$，证明了存在一个可数射影平面，它嵌入了所有秩为$3$的有限单拟阵，并且它的自同构群包含$Sym(ω)$，实际上我们证明了$Aut(F(M_*)) \cong Aut(M_*)$ .
英文摘要：
We construct a $\wedge$-homogeneous universal simple matroid of rank $3$, i.e. a countable simple rank~$3$ matroid $M_*$ which $\wedge$-embeds every finite simple rank $3$ matroid, and such that every isomorphism between finite $\wedge$-subgeometries of $M_*$ extends to an automorphism of $M_*$. We also construct a $\wedge$-homogeneous matroid $M_*(P)$ which is universal for the class of finite simple rank $3$ matroids omitting a given finite projective plane $P$. We then prove that these structures are not $\aleph_0$-categorical, they have the independence property, they admit a stationary independence relation, and that their automorphism group embeds the symmetric group $Sym(ω)$. Finally, we use the free projective extension $F(M_*)$ of $M_*$ to conclude the existence of a countable projective plane embedding all the finite simple matroids of rank $3$ and whose automorphism group contains $Sym(ω)$, in fact we show that $Aut(F(M_*)) \cong Aut(M_*)$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
