11 research outputs found
Universal homogeneous causal sets
Causal sets are particular partially ordered sets which have been proposed as
a basic model for discrete space-time in quantum gravity. We show that the
class C of all countable past-finite causal sets contains a unique causal set
(U,<) which is universal (i.e., any member of C can be embedded into (U,<)) and
homogeneous (i.e., (U,<) has maximal degree of symmetry). Moreover, (U,<) can
be constructed both probabilistically and explicitly. In contrast, the larger
class of all countable causal sets does not contain a universal object.Comment: 14 page
Invariant measures concentrated on countable structures
Let L be a countable language. We say that a countable infinite L-structure M
admits an invariant measure when there is a probability measure on the space of
L-structures with the same underlying set as M that is invariant under
permutations of that set, and that assigns measure one to the isomorphism class
of M. We show that M admits an invariant measure if and only if it has trivial
definable closure, i.e., the pointwise stabilizer in Aut(M) of an arbitrary
finite tuple of M fixes no additional points. When M is a Fraisse limit in a
relational language, this amounts to requiring that the age of M have strong
amalgamation. Our results give rise to new instances of structures that admit
invariant measures and structures that do not.Comment: 46 pages, 2 figures. Small changes following referee suggestion
Properties of the automorphism group and a probabilistic construction of a class of countable labeled structures
AbstractFor a class of countably infinite ultrahomogeneous structures that generalize edge-colored graphs we provide a probabilistic construction. Also, we give fairly general criteria for the automorphism group of such structures to have the small index property and strong uncountable cofinality, thus generalizing some results of Solecki, Rosendal, and several other authors
Multicoloured Random Graphs: Constructions and Symmetry
This is a research monograph on constructions of and group actions on
countable homogeneous graphs, concentrating particularly on the simple random
graph and its edge-coloured variants. We study various aspects of the graphs,
but the emphasis is on understanding those groups that are supported by these
graphs together with links with other structures such as lattices, topologies
and filters, rings and algebras, metric spaces, sets and models, Moufang loops
and monoids. The large amount of background material included serves as an
introduction to the theories that are used to produce the new results. The
large number of references should help in making this a resource for anyone
interested in beginning research in this or allied fields.Comment: Index added in v2. This is the first of 3 documents; the other 2 will
appear in physic