Distributivity of a Segmentation Lattice Jan Pavl k Brno - - PowerPoint PPT Presentation

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

1

is closed under all intersections

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

1

is closed under all intersections

2

contains all singletons

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

1

is closed under all intersections

2

contains all singletons

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

1

is closed under all intersections

2

contains all singletons

Since the second property is equivalent to T1 separation axiom, such a space will be referred to as a T1 closure space

• r 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

1

is closed under all intersections

2

contains all singletons

Since the second property is equivalent to T1 separation axiom, such a space will be referred to as a T1 closure space

• r 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

1

is closed under all intersections

2

contains all singletons

Since the second property is equivalent to T1 separation axiom, such a space will be referred to as a T1 closure space

• r 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.

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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 Part(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 Part(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 Part(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 Mi ⊆ X, i ∈ I is conditionally satisfied if p is satisfied whenever Mi ∩ Mj = ∅ 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 Mi ⊆ X, i ∈ I is conditionally satisfied if p is satisfied whenever Mi ∩ Mj = ∅ for all i, j ∈ I. For instance, ternary acyclicity is equivalent to conditional collective incidence (existence of a common point) for each triple

• f sets.

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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)

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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 ∨.

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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

• n finite spaces, but not necessary as shown bellow:

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Example: 4 element space

Let A = {a, b, c, d} and C contains, beside A and the singletons,

• nly the sets {a, c}, {b, c}. Then (A, C) is not connective since

{a, b, c} = {a, c} ∪ {b, c} is not closed but it is a union of two intersecting sets. However, the lattice S(C) is distributive. Indeed, since all non-singleton closed sets include the element c, every segmentation α ∈ S(C) is of the form λ(M) for some M ∈ C. Thus the Hasse diagrams for the lattices C and S(C) are:

∅ {a, c} {a} {c} {b} {b, c} A ∆ λ({a, c}) λ({b, c}) ∇

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Distinguished lattices

Recall the lattices M3 and N5 as shown bellow. Beside these we introduce two more ordered sets S6 and K7: a b c 1 a b c 1 a ab b bc c 1 a ab ac b bc c M3 N5 S6 K7

Motivation SD-spaces Examples Characterization of SD-spaces by embeddings

Characterization of segmentationally distributive spaces

Theorem A space (X, C) is segmentationally distributive iff it satisfies the following conditions: P1 j(0) = ∅ for every injective ∧-preserving map j : K7 → C,

Motivation SD-spaces Examples Characterization of SD-spaces by embeddings

Characterization of segmentationally distributive spaces

Theorem A space (X, C) is segmentationally distributive iff it satisfies the following conditions: P1 j(0) = ∅ for every injective ∧-preserving map j : K7 → C, P2 j(0) = ∅ for every injective lattice homomorphism j : N5 → C,

Motivation SD-spaces Examples Characterization of SD-spaces by embeddings

Characterization of segmentationally distributive spaces

Theorem A space (X, C) is segmentationally distributive iff it satisfies the following conditions: P1 j(0) = ∅ for every injective ∧-preserving map j : K7 → C, P2 j(0) = ∅ for every injective lattice homomorphism j : N5 → C, P3 j(0) = ∅ for every injective lattice homomorphism j : M3 → C,

Motivation SD-spaces Examples Characterization of SD-spaces by embeddings

Characterization of segmentationally distributive spaces

Theorem A space (X, C) is segmentationally distributive iff it satisfies the following conditions: P1 j(0) = ∅ for every injective ∧-preserving map j : K7 → C, P2 j(0) = ∅ for every injective lattice homomorphism j : N5 → C, P3 j(0) = ∅ for every injective lattice homomorphism j : M3 → C, P4 There is no injective lattice homomorphism j : S6 → C.

Motivation SD-spaces Examples Characterization of SD-spaces by embeddings

Proof of the theorem: Outline

Outline of the proof: (D1)⇔(P1) ((D1)&(D2))⇒((P2)&(P3)) ((P2)&(P3))⇒(D2) (D3)⇒(P4) ((D1)&(D2)&(P4))⇒(D3)

Motivation SD-spaces Examples Characterization of SD-spaces by embeddings

Proof of the theorem: (D1)⇔(P1)

The ternary acyclicity, i.e., (M ∩ N = ∅ & M ∩ P = ∅ & N ∩ P = ∅) ⇒ M ∩ N ∩ P = ∅ is equivalent to

Motivation SD-spaces Examples Characterization of SD-spaces by embeddings

Proof of the theorem: (D1)⇔(P1)

The ternary acyclicity, i.e., (M ∩ N = ∅ & M ∩ P = ∅ & N ∩ P = ∅) ⇒ M ∩ N ∩ P = ∅ is equivalent to j(0) = ∅ for every injective ∧-preserving map j : a ab ac b bc c → C.

Motivation SD-spaces Examples Characterization of SD-spaces by embeddings

Proof of the theorem: ((D1)&(D2))⇒((P2)&(P3))

The ternary acyclicity together with and the conditional distributivity of ∩, i.e.,

(M∩N = ∅ & M∩P = ∅ & N∩P = ∅) ⇒ (M∩P)∨(N∩P) = (M∨N)∩P

imply

Motivation SD-spaces Examples Characterization of SD-spaces by embeddings

Proof of the theorem: ((D1)&(D2))⇒((P2)&(P3))

The ternary acyclicity together with and the conditional distributivity of ∩, i.e.,

(M∩N = ∅ & M∩P = ∅ & N∩P = ∅) ⇒ (M∩P)∨(N∩P) = (M∨N)∩P

imply j(0) = ∅ for every injective ∧-preserving map j a b c 1 → C

• r

a b c 1 → C.

Motivation SD-spaces Examples Characterization of SD-spaces by embeddings

Proof of the theorem: ((P2)&(P3))⇒(D2)

The conditional distributivity of ∩ follows from

Motivation SD-spaces Examples Characterization of SD-spaces by embeddings

Proof of the theorem: ((P2)&(P3))⇒(D2)

The conditional distributivity of ∩ follows from the condition that j(0) = ∅ for every injective ∧-preserving map j M3 → C or N5 → C.

Motivation SD-spaces Examples Characterization of SD-spaces by embeddings

Proof of the theorem: (D3)⇒(P4)

The weakly conditional distributivity of ∨, i.e., (M ∩ P = ∅ & N ∩ P = ∅) ⇒ (M ∨ P) ∩ (N ∨ P) = (M ∩ N) ∨ P implies

Motivation SD-spaces Examples Characterization of SD-spaces by embeddings

Proof of the theorem: (D3)⇒(P4)

The weakly conditional distributivity of ∨, i.e., (M ∩ P = ∅ & N ∩ P = ∅) ⇒ (M ∨ P) ∩ (N ∨ P) = (M ∩ N) ∨ P implies non-existence of a lattice homomorphism j : a ab b bc c 1 → C.

Motivation SD-spaces Examples Characterization of SD-spaces by embeddings

Proof of the theorem: ((D1)&(D2)&(P4))⇒(D3)

The ternary acyclicity and the conditional distributivity of ∩ and the non-existence of an injective lattice homomorphism j : S6 → C implies the weakly conditional distributivity of ∨.

Motivation SD-spaces Examples Characterization of SD-spaces by embeddings

Thank you for your attention.