Motivation SD-spaces Examples Characterization of SD-spaces by embeddings Distributivity of a Segmentation Lattice Jan Pavl´ ık Brno University of Technology Brno, Czech Republic
Motivation SD-spaces Examples Characterization of SD-spaces by embeddings The problem The problem is simple stated. Problem Characterize all segmentationally distributive T1 closure spaces.
Motivation SD-spaces Examples Characterization of SD-spaces by embeddings The problem The problem is simple stated. Problem Characterize all segmentationally distributive T1 closure spaces. We will focus on finitary spaces only.
Motivation SD-spaces Examples Characterization of SD-spaces by embeddings T1 C-spaces What does it mean? A closure space with closed singletons is a set ( the base set ) together with a system of subsets ( closed sets ) which
Motivation SD-spaces Examples Characterization of SD-spaces by embeddings T1 C-spaces What does it mean? A closure space with closed singletons is a set ( the base set ) together with a system of subsets ( closed sets ) which is closed under all intersections 1
Motivation SD-spaces Examples Characterization of SD-spaces by embeddings T1 C-spaces What does it mean? A closure space with closed singletons is a set ( the base set ) together with a system of subsets ( closed sets ) which is closed under all intersections 1 contains all singletons 2
Motivation SD-spaces Examples Characterization of SD-spaces by embeddings T1 C-spaces What does it mean? A closure space with closed singletons is a set ( the base set ) together with a system of subsets ( closed sets ) which is closed under all intersections 1 contains all singletons 2
Motivation SD-spaces Examples Characterization of SD-spaces by embeddings T1 C-spaces What does it mean? A closure space with closed singletons is a set ( the base set ) together with a system of subsets ( closed sets ) which is closed under all intersections 1 contains all singletons 2 Since the second property is equivalent to T1 separation axiom, such a space will be referred to as a T1 closure space or briefly a T1 C-space .
Motivation SD-spaces Examples Characterization of SD-spaces by embeddings T1 C-spaces What does it mean? A closure space with closed singletons is a set ( the base set ) together with a system of subsets ( closed sets ) which is closed under all intersections 1 contains all singletons 2 Since the second property is equivalent to T1 separation axiom, such a space will be referred to as a T1 closure space or briefly a T1 C-space . A segmentation of a given T1 C-space is a partition of the base set into the closed sets.
Motivation SD-spaces Examples Characterization of SD-spaces by embeddings T1 C-spaces What does it mean? A closure space with closed singletons is a set ( the base set ) together with a system of subsets ( closed sets ) which is closed under all intersections 1 contains all singletons 2 Since the second property is equivalent to T1 separation axiom, such a space will be referred to as a T1 closure space or briefly a T1 C-space . A segmentation of a given T1 C-space is a partition of the base set into the closed sets. A C-space is finitary if closure of each set is the union of closures of its finite subsets.
Motivation SD-spaces Examples Characterization of SD-spaces by embeddings The segmenatation lattice Let ( X , C ) be a T1 C-space. Canonical segmentation Each M ∈ C induces a segmentation λ ( M ) whose only possible nontrivial class is M . Clearly λ ( { x } ) = ∆ for each x ∈ X and λ ( X ) = ∇ .
Motivation SD-spaces Examples Characterization of SD-spaces by embeddings The segmenatation lattice Let ( X , C ) be a T1 C-space. Canonical segmentation Each M ∈ C induces a segmentation λ ( M ) whose only possible nontrivial class is M . Clearly λ ( { x } ) = ∆ for each x ∈ X and λ ( X ) = ∇ . Proposition The set of all segmentations S ( C ) forms a complete lattice which is a sub- ∧ -semilattice of the lattice P art ( X ) of all partitions on X.
Motivation SD-spaces Examples Characterization of SD-spaces by embeddings The segmenatation lattice Let ( X , C ) be a T1 C-space. Canonical segmentation Each M ∈ C induces a segmentation λ ( M ) whose only possible nontrivial class is M . Clearly λ ( { x } ) = ∆ for each x ∈ X and λ ( X ) = ∇ . Proposition The set of all segmentations S ( C ) forms a complete lattice which is a sub- ∧ -semilattice of the lattice P art ( X ) of all partitions on X. Definition ( X , C ) is called distributive if C is distributive,
Motivation SD-spaces Examples Characterization of SD-spaces by embeddings The segmenatation lattice Let ( X , C ) be a T1 C-space. Canonical segmentation Each M ∈ C induces a segmentation λ ( M ) whose only possible nontrivial class is M . Clearly λ ( { x } ) = ∆ for each x ∈ X and λ ( X ) = ∇ . Proposition The set of all segmentations S ( C ) forms a complete lattice which is a sub- ∧ -semilattice of the lattice P art ( X ) of all partitions on X. Definition ( X , C ) is called distributive if C is distributive, segmentationally distributive (an SD-space ) if S ( C ) is distributive.
Motivation SD-spaces Examples Characterization of SD-spaces by embeddings The motivation Why do we ask that?
Motivation SD-spaces Examples Characterization of SD-spaces by embeddings The motivation Why do we ask that? Inner structure on a T1 C-space Given a T1 C-space ( X , C ), a there is a canonical generalized ultrametric (GUM) ω on A with values in S ( C ) compatible with C in such a way that closed sets coincide with closed balls w.r.t. ω .
Motivation SD-spaces Examples Characterization of SD-spaces by embeddings The motivation Why do we ask that? Inner structure on a T1 C-space Given a T1 C-space ( X , C ), a there is a canonical generalized ultrametric (GUM) ω on A with values in S ( C ) compatible with C in such a way that closed sets coincide with closed balls w.r.t. ω . Application of a distributive lattice valued GUM spaces Given a GUM spaces X with values in a complete lattice L , then there exists an action of L on X . If L is distributive lattice, then there is an adjoint operation to the scalar action (a scalar division) which enables to construct an algorithm for finding a segmentation based upon connection of pairs of elements.
Motivation SD-spaces Examples Characterization of SD-spaces by embeddings Auxiliary properties Ternary acyclicity The property for sets M , N , P ⊆ X ( M ∩ N � = ∅ & M ∩ P � = ∅ & N ∩ P � = ∅ ) ⇒ M ∩ N ∩ P � = ∅ (D1) will be called ternary acyclicity .
Motivation SD-spaces Examples Characterization of SD-spaces by embeddings Auxiliary properties Ternary acyclicity The property for sets M , N , P ⊆ X ( M ∩ N � = ∅ & M ∩ P � = ∅ & N ∩ P � = ∅ ) ⇒ M ∩ N ∩ P � = ∅ (D1) will be called ternary acyclicity . Moreover we define: Definition A property p on sets M i ⊆ X , i ∈ I is conditionally satisfied if p is satisfied whenever M i ∩ M j � = ∅ for all i , j ∈ I .
Motivation SD-spaces Examples Characterization of SD-spaces by embeddings Auxiliary properties Ternary acyclicity The property for sets M , N , P ⊆ X ( M ∩ N � = ∅ & M ∩ P � = ∅ & N ∩ P � = ∅ ) ⇒ M ∩ N ∩ P � = ∅ (D1) will be called ternary acyclicity . Moreover we define: Definition A property p on sets M i ⊆ X , i ∈ I is conditionally satisfied if p is satisfied whenever M i ∩ M j � = ∅ for all i , j ∈ I . For instance, ternary acyclicity is equivalent to conditional collective incidence (existence of a common point) for each triple of sets.
Motivation SD-spaces Examples Characterization of SD-spaces by embeddings Conditional distributivity Namely we will use the notion of conditional distributivity of ∩ (D2). Considering a T1 C-space ( X , C ), by a slight abuse of language we also consider the weakly conditional distributivity of ∨ in the lattice C : for all M , N , P ∈ C ( M ∩ P � = ∅ & N ∩ P � = ∅ ) ⇒ ( M ∨ P ) ∩ ( N ∨ P ) = ( M ∩ N ) ∨ P , (D3)
Motivation SD-spaces Examples Characterization of SD-spaces by embeddings Characterization theorem Theorem A finitary segmentation space ( X , C ) is segmentationally distributive iff for each three closed sets M , N , P ∈ C is satisfied the conjunction of the conditions: 1 the ternary acyclicity, 2 the conditional distributivity of ∩ , 3 the weakly conditional distributivity of ∨ .
Motivation SD-spaces Examples Characterization of SD-spaces by embeddings Example: Tree space A tree graph is a ternary acyclic closure space if we consider the connected sets to be closed. It is a segmentationally distributive space since the segmentations of the tree are in one-to-one correspondence with the subsets of set of edges.
Motivation SD-spaces Examples Characterization of SD-spaces by embeddings Example: Tree space A tree graph is a ternary acyclic closure space if we consider the connected sets to be closed. It is a segmentationally distributive space since the segmentations of the tree are in one-to-one correspondence with the subsets of set of edges. This space is a connective space - the supremum of incident pair of closed sets is their union. It can be shown that this property together with ternary acyclicity is sufficient for the segmentational distributivity on finite spaces, but not necessary as shown bellow:
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