We construct a Lambda-homogeneous universal simple matroid of rank 3, i.e. a countable simple rank 3 matroid M* which Lambda-embeds every finite simple rank 3 matroid, and such that every isomorphism between finite Lambda-subgeometries of M* extends to an automorphism of M*. We also construct a Lambda-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 N-0-categorical, they have the independence property, they admit a stationary independence relation, and that their automorphism group embeds the symmetric group Sym(omega). 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(omega), in fact we show that Aut(F(M*)) congruent to Aut(M*).
A Universal Homogeneous Simple Matroid of Rank 3
Paolini, G
2018-01-01
Abstract
We construct a Lambda-homogeneous universal simple matroid of rank 3, i.e. a countable simple rank 3 matroid M* which Lambda-embeds every finite simple rank 3 matroid, and such that every isomorphism between finite Lambda-subgeometries of M* extends to an automorphism of M*. We also construct a Lambda-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 N-0-categorical, they have the independence property, they admit a stationary independence relation, and that their automorphism group embeds the symmetric group Sym(omega). 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(omega), in fact we show that Aut(F(M*)) congruent to Aut(M*).
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