An explicit construction for large sets of infinite dimensional qq-Steiner systems

Let $V$ be a vector space over the finite field ${\mathbb F}_q$. A $q$-Steiner system, or an $S(t,k,V)_q$, is a collection ${\mathcal B}$ of $k$-dimensional subspaces of $V$ such that every $t$-dimensional subspace of $V$ is contained in a unique element of ${\mathcal B}$. A large set of $q$-Steiner systems, or an $LS(t,k,V)_q$, is a partition of the $k$-dimensional subspaces of $V$ into $S(t,k,V)_q$ systems. In the case that $V$ has infinite dimension, the existence of an $LS(t,k,V)_q$ for all finite $t,k$ with $1<t<k$ was shown by Cameron in 1995. This paper provides an explicit construction of an $LS(t,t+1,V)_q$ for all prime powers $q$, all positive integers $t$, and where $V$ has countably infinite dimension.Comment: 5 page

Some examples related to Conway Groupoids

We discuss the recently introduced notion of a Conway Groupoid. In particular we consider various generalisations of the concept including infinite analogues

Resolvability of infinite designs

In this paper we examine the resolvability of infinite designs. We show that in stark contrast to the finite case, resolvability for infinite designs is fairly commonplace. We prove that every t-(v,k,Λ) design with t finite, v infinite and k,λ<v is resolvable and, in fact, has α orthogonal resolutions for each α<v. We also show that, while a t-(v,k,Λ) design with t and λ finite, v infinite and k=v may or may not have a resolution, any resolution of such a design must have v parallel classes containing v blocks and at most λ−1 parallel classes containing fewer than v blocks. Further, a resolution into parallel classes of any specified sizes obeying these conditions is realisable in some design. When k<v and λ=v and when k=v and λ is infinite, we give various examples of resolvable and non-resolvable t-(v,k,Λ) designs

Some examples related to Conway Groupoids and their generalisations

We discuss the recently introduced notion of a Conway Groupoid. In particular we consider various generalisations of the concept including infinite analogues

Infinite Jordan Permutation Groups

Abstract If G is a transitive permutation group on a set X, then G is a Jordan group if there is a partition of X into non-empty subsets Y and Z with |Z| > 1, such that the pointwise stabilizer in G of Y acts transitively on Z (plus other non-degeneracy conditions). There is a classification theorem by Adeleke and Macpherson for the infinite primitive Jordan permutation groups: such group preserves linear-like structures, or tree-like structures, or Steiner systems or a ‘limit’ of Steiner systems, or a ‘limit’ of betweenness relations or D-relations. In this thesis we build a structure M whose automorphism group is an infinite oligomorphic primitive Jordan permutation group preserving a limit of D-relations. In Chapter 2 we build a class of finite structures, each of which is essentially a finite lower semilinear order with vertices labelled by finite D-sets, with coherence conditions. These are viewed as structures in a relational language with relations L,L',S,S',Q,R. We describe possible one point extensions, and prove an amalgamation theorem. We obtain by Fra¨ıss´e’s Theorem a Fra¨ıss´e limit M. In Chapter 3, we describe in detail the structure M and its automorphism group. We show that there is an associated dense lower semilinear order, again with vertices labelled by (dense) D-sets, again with coherence conditions. By a method of building an iterated wreath product described by Cameron which is based on Hall’s wreath power, we build in Chapter 4 a group K < Aut(M) which is a Jordan group with a pre-direction as its Jordan set. Then we find, by properties of Jordan sets, that a pre-D-set is a Jordan set for Aut(M). Finally we prove that the Jordan group G = Aut(M) preserves a limit of D-relations as a main result of this thesis