14,586 research outputs found
The connection matrix theory for semiflows on (not necessarily locally compact) metric spaces
AbstractThe index theory of Rybakowski for isolated invariant sets and attractor-repeller pairs in the setting of a semiflow on a not necessarily locally compact metric space is extended to include a connection matrix theory for Morse decompositions. Partially ordered Morse decompositions and attractor semifiltrations of invariant sets are defined and shown to be equivalent. The definition and proof of existence of index filtrations for an ordered Morse decomposition is provided. Via the index filtration, the homology index braid and the connection matrices of the Morse decomposition are defined
Decompositions of Modules Associated to Finite Partially Ordered Sets
AbstractFor a finite partially ordered set L and a field F, let F L be the associated vector space with L as basis. In the case of ranked posets this space decomposes into eigenspaces under the maps afforded by the order relation. In this note we show how to construct generating sets for such decompositions and discuss their combinatorial significance
Type-Decomposition of a Pseudo-Effect Algebra
The theory of direct decomposition of a centrally orthocomplete effect
algebra into direct summands of various types utilizes the notion of a
type-determining (TD) set. A pseudo-effect algebra (PEA) is a (possibly)
noncommutative version of an effect algebra. In this article we develop the
basic theory of centrally orthocomplete PEAs, generalize the notion of a TD set
to PEAs, and show that TD sets induce decompositions of centrally orthocomplete
PEAs into direct summands.Comment: 18 page
Measurable realizations of abstract systems of congruences
An abstract system of congruences describes a way of partitioning a space
into finitely many pieces satisfying certain congruence relations. Examples of
abstract systems of congruences include paradoxical decompositions and
-divisibility of actions. We consider the general question of when there are
realizations of abstract systems of congruences satisfying various
measurability constraints. We completely characterize which abstract systems of
congruences can be realized by nonmeager Baire measurable pieces of the sphere
under the action of rotations on the -sphere. This answers a question of
Wagon. We also construct Borel realizations of abstract systems of congruences
for the action of on .
The combinatorial underpinnings of our proof are certain types of decomposition
of Borel graphs into paths. We also use these decompositions to obtain some
results about measurable unfriendly colorings.Comment: minor correction
- …