6 research outputs found
A Universal Homogeneous Simple Matroid of Rank
We construct a -homogeneous universal simple matroid of rank ,
i.e. a countable simple rank~ matroid which -embeds every
finite simple rank matroid, and such that every isomorphism between finite
-subgeometries of extends to an automorphism of . We also
construct a -homogeneous matroid which is universal for the
class of finite simple rank matroids omitting a given finite projective
plane . We then prove that these structures are not -categorical,
they have the independence property, they admit a stationary independence
relation, and that their automorphism group embeds the symmetric group
. Finally, we use the free projective extension of
to conclude the existence of a countable projective plane embedding all the
finite simple matroids of rank and whose automorphism group contains
, in fact we show that
A classification of finite homogeneous semilinear spaces
Abstract. A semilinear space S is homogeneous if, whenever the semilinear structures induced on two finite subsets S1 and S2 of S are isomorphic, there is at least one automorphism of S mapping S1 onto S2. We give a complete classification of all finite homogeneous semilinear spaces. Our theorem extends a result of Ronse on graphs and a result of Devillers and Doyen on linear spaces. Key words. Semilinear space, polar space, copolar space, partial geometry, automorphism group, homogeneity. 2000 Mathematics Subject Classification. 05B25, 51E14, 20B25
Countable homogeneous Steiner triple systems avoiding specified subsystems
In this article we construct uncountably many new homogeneous locally finite Steiner triple systems of countably infinite order as Fraïssé limits of classes of finite Steiner triple systems avoiding certain subsystems. The construction relies on a new embedding result: any finite partial Steiner triple system has an embedding into a finite Steiner triple system that contains no nontrivial proper subsystems that are not subsystems of the original partial system. Fraïssé’s construction and its variants are rich sources of examples that are central to model-theoretic classification theory, and recently infinite Steiner systems obtained via Fraïssé-type constructions have received attention from the model theory community
Homogeneous and Ultrahomogeneous Linear Spaces
A linear spaceSishomogeneousif, whenever the linear structures induced on two finite subsetsS′ andS″ are isomorphic, there is at least one automorphism ofSmappingS′ ontoS″. If every isomorphism fromS′ toS″ can be extended to an automorphism ofS,Sis calledultrahomogeneous. We give a complete classification of all homogeneous (resp. ultrahomogeneous) linear spaces, without making any finiteness assumption on the number of points ofS. © 1998 Academic Press.SCOPUS: ar.jinfo:eu-repo/semantics/publishe