149 research outputs found
Some more notions of homomorphism-homogeneity
We extend the notion of 'homomorphism-homogeneity' to a wider class of kinds
of maps than previously studied, and we investigate the relations between the
resulting notions of homomorphism-homogeneity, giving several examples. We also
give further details on related work reported in [Deborah Lockett and John K
Truss, Generic endomorphisms of homogeneous structures, in 'Groups and model
theory', Contemporary Mathematics 576, ed Strungmann, Droste, Fuchs, Tent,
American Mathematical Society, 2012, 217-237] about the endomorphisms of chains
and generic endomorphisms of tree.Comment: 15 page
On automatic homeomorphicity for transformation monoids
Transformation monoids carry a canonical topology --- the topology of
point-wise convergence. A closed transformation monoid is said
to have automatic homeomorphicity with respect to a class of
structures, if every monoid-isomorphism of to the endomorphism
monoid of a member of is automatically a homeomorphism. In this
paper we show automatic homeomorphicity-properties for the monoid of
non-decreasing functions on the rationals, the monoid of non-expansive
functions on the Urysohn space and the endomorphism-monoid of the countable
universal homogeneous poset.Comment: 21 page
A universality result for endomorphism monoids of some ultrahomogeneous structures
We devise a fairly general sufficient condition ensuring that the
endomorphism monoid of a countably infinite ultrahomogeneous structure (i.e. a
Fra\"{\i}ss\'{e} limit) embeds all countable semigroups. This approach provides
us not only with a framework unifying the previous scattered results in this
vein, but actually yields new applications for endomorphism monoids of the
(rational) Urysohn space and the countable universal ultrahomogeneous
semilattice.Comment: 20 pages; to appear in the Proceedings of the Edinburgh Mathematical
Societ
The Bergman property for endomorphism monoids of some Fra\"{\i}ss\'e limits
Based on an idea of Y. P\'eresse and some results of Maltcev, Mitchell and
Ru\v{s}kuc, we present sufficient conditions under which the endomorphism
monoid of a countably infinite ultrahomogeneous first-order structure has the
Bergman property. This property has played a prominent role both in the theory
of infinite permutation groups and, more recently, in semigroup theory. As a
byproduct of our considerations, we establish a criterion for a countably
infinite ultrahomogeneous structure to be homomorphism-homogeneous.Comment: 16 pages; to appear in Forum Mathematicu
Universal homomorphisms, universal structures, and the polymorphism clones of homogeneous structures
Using a categorial version of Fra\"iss\'e's theorem due to Droste and
G\"obel, we derive a criterion for a comma-category to have universal
homogeneous objects.
As a first application we give new existence result for universal structures
and for \omega-categorical universal structures.
As a second application we characterize the retracts of a large class of
homogeneous structures, extending previous results by Bonato, Deli\'c, Dolinka,
and Kubi\'s.
As a third application we show for a large class of homogeneous structures
that their polymorphism clone is generated by polymorphisms of bounded arity,
generalizing a classical result by Sierpi\'nski that the clone of all functions
on a given set is generated by its binary part.
Further we study the cofinality and the Bergman property for clones and we
give sufficient conditions on a homogeneous structure to have a polymorphism
clone that has uncountable cofinality and the Bergman property.Comment: corrected several typo
Universality of intervals of line graph order
We prove that for every the homomorphism order of the class of line
graphs of finite graphs with maximal degree is universal. This means that
every finite or countably infinite partially ordered set may be represented by
line graphs of graphs with maximal degree ordered by the existence of a
homomorphism.Comment: 13 pages, 8 figures, accepted to European Journal of Combinatoric
Polymorphism clones of homogeneous structures (Universal homogeneous polymorphisms and automatic homeomorphicity)
Every clone of functions comes naturally equipped with a topology---the
topology of pointwise convergence. A clone is said to have
automatic homeomorphicity with respect to a class of clones, if
every clone-isomorphism of to a member of is
already a homeomorphism (with respect to the topology of pointwise
convergence). In this paper we study automatic homeomorphicity-properties for
polymorphism clones of countable homogeneous relational structures. To this end
we introduce and utilize universal homogeneous polymorphisms. Next to two
generic criteria for the automatic homeomorphicity of the polymorphism clones
of free homogeneous structures we show that the polymorphism clone of the
generic poset with reflexive ordering has automatic homeomorphicity and that
the polymorphism clone of the generic poset with strict ordering has automatic
homeomorphicity with respect to countable -categorical structures. Our
results extend and generalize previous results by Bodirsky, Pinsker, and
Pongr\'acz.Comment: 38 pages, revised and extended versio
D-Ultrafilters and their Monads
For a number of locally finitely presentable categories K we describe the
codensity monad of the full embedding of all finitely presentable objects into
K. We introduce the concept of D-ultrafilter on an object, where D is a "nice"
cogenerator of K. We prove that the codensity monad assigns to every object an
object representing all D-ultrafilters on it. Our result covers e.g. categories
of sets, vector spaces, posets, semilattices, graphs and M-sets for finite
commutative monoids M.Comment: 25 page
Homomorphism-homogeneous L-colored graphs
A relational structure is homomorphism-homogeneous (HH-homogeneous for short)
if every homomorphism between finite induced substructures of the structure can
be extended to a homomorphism over the whole domain of the structure.
Similarly, a structure is monomorphism-homogeneous (MH-homogeneous for short)
if every monomorphism between finite induced substructures of the structure can
be extended to a homomorphism over the whole domain of the structure. In this
paper we consider L-colored graphs, that is, undirected graphs without loops
where sets of colors selected from L are assigned to vertices and edges. A full
classification of finite MH-homogeneous L-colored graphs where L is a chain is
provided, and we show that the classes MH and HH coincide. When L is a diamond,
that is, a set of pairwise incomparable elements enriched with a greatest and a
least element, the situation turns out to be much more involved. We show that
in the general case the classes MH and HH do not coincide.Comment: Submitted to European Journal of Combinatoric
Two Fra\"iss\'e-style theorems for homomorphism-homogeneous relational structures
In this paper, we state and prove two Fra\"{i}ss\'{e}-style results that
cover existence and uniqueness properties for twelve of the eighteen different
notions of homomorphism-homogeneity as introduced by Lockett and Truss, and
provide forward directions and implications for the remaining six cases.
Following these results, we completely determine the extent to which the
countable homogeneous undirected graphs (as classified by Lachlan and Woodrow)
are homomorphism-homogeneous; we also provide some insight into the directed
graph case.Comment: 28 pages, 12 figures, 2 table
- …