The Scaffolding of a Formal Context Stephan Doerfel 1 , 2 1 Knowledge - - PowerPoint PPT Presentation

The Scaffolding of a Formal Context

Stephan Doerfel1,2

1Knowledge and Data Engineering Group, Department of Electrical Engineering and

Computer Science, University of Kassel

2Department of Mathematics, Institute of Algebra,

Technical University of Dresden

Concept Lattices and Their Applications - CLA 2010 October 20th, 2010

Stephan Doerfel (University Kassel) Scaffolding CLA 2010 1 / 31

Agenda

1

Motivation

2

The Scaffolding of a Lattice

3

The Scaffolding of a Formal Context

4

Example and Diagram

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Agenda

1

Motivation

2

The Scaffolding of a Lattice

3

The Scaffolding of a Formal Context

4

Example and Diagram

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Motivation

Small contexts can have very large lattices Enumerating all lattice elements is expensive Diagrams - though easy to interpret - can become hard to read Rudolf Wille constructed a smaller representation of a complete lattice of finite length: The Scaffolding

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Example - Lattice

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Example - Scaffolding

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Idea

What does the scaffolding look like in the world of formal contexts? How can it be constructed using the means of FCA? How can it be constructed without constructing the whole concept lattice first?

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Agenda

1

Motivation

2

The Scaffolding of a Lattice

3

The Scaffolding of a Formal Context

4

Example and Diagram

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History

Definition and construction of the scaffolding by R. Wille (1976) in ”‘Subdirekte Produkte vollst¨ andiger Verb¨ ande”’ Picked up by B. Ganter, W. Poguntke und R. Wille (1981) in ”‘Finite sublattices of four-generated modular lattices“’ Related: ”The core of finite lattices“ by V. Duquenne

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Construction means 1/2

Residual Map

For complete lattices L1 and L2 and a surjective inf-morphism α : L1 → L2, the map α : L2 → L1, x → inf α−1x is an injective sup-morphism, called the residual map to α.

Separating Maps

A set of maps αt : L → Lt (t ∈ T) is called separating if for all x = y ∈ L there is an index t such that αt(x) = αt(y) .

Stephan Doerfel (University Kassel) Scaffolding CLA 2010 10 / 31

Construction means 2/2

Relative sup-subsemilattice

A relative sup-subsemilattice of a complete lattice L is a subset S ⊆ L together with a partial operation supS such that supS A = s ⇐ ⇒ sup A = s holds for A ⊆ S and s ∈ S.

Theorem

For an arbitrary index set T, complete lattices L and Lt (t ∈ T) and separating complete homomorphisms αt : L → Lt S(αt | t ∈ T) := {αtαtx | x ∈ L, t ∈ T} \ {0} is a supremum-dense subset of L and L is isomorphic to the complete lattice of complete ideals of the relative sup-subsemilattice S(αt | t ∈ T).

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Scaffolding 2/2

Scaffolding

The scaffolding of a complete lattice L of finite length is S(L) := S(αt | t ∈ T) = {αtαtx | x ∈ L, t ∈ T} \ {0} where Lt are all completely subdirectly irreducible factors of L αt are all surjective complete homomorphisms αt : L → Lt

Subirreducible elements

An element x ∈ L is called subirreducible if it is an element of S(L), i. e. if a complete homomorphism α from L onto a subdirectly irreducible factor of L exists with ααx = x.

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Agenda

1

Motivation

2

The Scaffolding of a Lattice

3

The Scaffolding of a Formal Context

4

Example and Diagram

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Means

Now, given a formal context, we need equivalents for separating morphisms residuals subdirectly irreducible factors subirreducible elements

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Bonds

Kx = (Gx, Mx, Ix)

Bond

A bond from Ks to Kt is a relation Rst ⊆ Gs × Mt, such that gRst is an intent of Kt for every object g ∈ Gs and mRst is an extent of Ks for every attribute m ∈ Mt.

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Bonds and morphisms

Ms Mt Gs Ks Rst Gt Rts Kt α : (A, B) → (ARstIt, ARst) is a sup-morphism B(Ks) → B(Kt) α : (A, B) → (BRst, BRstIs) is a inf-morphism B(Kt) → B(Ks) α is residual to α α is a complete homomorphism iff a bond Rts exists such that ∀(A, B) ∈ B(Ks) holds ARstIt = BRts. Such a pair (Rst, Rts) we will call hom-bonds.

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Products of bonds

Product and Composition

For bonds Rrs from Kr to Ks and Rst from Ks to Kt the bond product is defined by Rrs ◦ Rst := {(g, m) ∈ Gr × Mt | gRrsIs ⊆ mRst} For the corresponding sup-morphisms holds: φRrs◦Rst = φRst ◦ φRrs.

Definition

A set of hom-bonds (Rt, St) between K und Kt is called separating if for any two extents A = C of K there is an index t ∈ T such that ARt = C Rt holds.

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Construction

supremum-dense subset

For separating hom-bonds (Rt, St) between K and Kt (t ∈ T) S(Rt, St)t∈T := {(ARt◦StI, ARt◦St) | (A, B) ∈ B(K), t ∈ T} \ {(MI, M)} is a supremum-dense subset of B(K).

Theorem

For contexts K and Kt, separating hom-bonds (Rt, St) between K and Kt and their corresponding homomorphisms αt : B(K) → B(Kt) holds S(αt | t ∈ T) = S(Rt, St)t∈T.

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Doubly Founded, Arrows

Doubly founded

A complete lattice L is called doubly founded, if for any two elements x < y ∈ L there always are elements s, t ∈ L with s being minimal w.r.t. s ≤ y and s x and t being maximal w.r.t. t ≥ x and t y.

Arrows

For a context (G, M, I) and g ∈ G, m ∈ M and (g, m) / ∈ I the arrow relations are defined as g ւ m : ⇐ ⇒ (∀h ∈ G : gI ⊆ hI, gI = hI = ⇒ (h, m) ∈ I), g ր m : ⇐ ⇒ (∀n ∈ M : mI ⊆ nI, mI = nI = ⇒ (g, n) ∈ I), g ր ւ m : ⇐ ⇒ g ւ m and g ր m.

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Arrow-closed subcontexts

Double Arrows

In a reduced context of a doubly founded lattice for every object g exists at least one attribute m with g ր ւ m. The analogue holds for every attribute.

Subcontexts

A subcontext (H, N, J) of a reduced context (G, M, I) is called arrow-closed, if for g ∈ G and m ∈ M it always holds that from g ∈ H and g ր m follows m ∈ N and from m ∈ N and g ւ m follows g ∈ H. For an object g ∈ G there always exists a smallest arrow-closed subcontext gK = (Gg, Mg, Ig) containing g, called the 1-generated subcontext of g in (G, M, I).

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The Scaffolding

Separating hom-bonds

If K is a reduced context of a doubly founded concept lattice and gK = (Gg, Mg, Ig), then (Rg, Sg) with Rg := I ∩ (G × Mg) and Sg := I ∩ (Gg × M) (g ∈ G) are separating hom-bonds.

Definition

If B(K) is the doubly founded concept lattice of a reduced context K and (Rg, Sg) are as above, then the relative sup-semilattice S(Rg, Sg)g∈G = {(ARg◦SgI, ARg◦Sg ) | (A, B) ∈ B(K), g ∈ G} \ {(MI, M)} is called the scaffolding of K and will be denoted by S(K).

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Results

Theorem

For a reduced context K of a lattice of finite length the scaffolding of the context S(K) is equal to the scaffolding of the lattice S(B(K)).

Subirreducible Elements

In a reduced context K of a doubly founded concept lattice a concept (A, B) with B = M is subirreducible, iff there is a 1-generated arrow-closed subcontext (H, N, J) such that (A, B) = ((A ∩ H)II, (A ∩ H)I) .

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Simplification

S(Rg, Sg)g∈G = {(ARg◦SgI, ARg◦Sg ) | (A, B) ∈ B(K), g ∈ G} \ {(MI, M)}

Simplified Scaffolding

For a reduced context K of a doubly founded concept lattice with 1-generated subcontexts Kt (t ∈ T) covering K holds S(K) =

• t∈T

{(C II, C I) | (C, C It) ∈ B(Kt), C It = Mt} . The sets {(C II, C I) | (C, C It) ∈ B(Kt), C It = Mt} (t ∈ T) are called the components of the scaffolding.

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Agenda

1

Motivation

2

The Scaffolding of a Lattice

3

The Scaffolding of a Formal Context

4

Example and Diagram

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5 1 6 2 4 3 7 g a e d f b

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A context

a b c d e f g 1 × ×

ր ւ

× × × × 2

ր ւ

×

ր

× × × × 3

ւ ր ւ

×

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× 4 × ×

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× × 5 × × × × × ×

ր ւ

6 × × × ×

ր ւ

×

ր

7 × ×

ւ ր ւ

×

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3 convering subcontexts

a b c d e f g 1 × ×

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× × × × 2

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×

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× × × × 3

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×

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× 4 × ×

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× × 5 × × × × × ×

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6 × × × ×

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7 × ×

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× a b c 1 × ×

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2

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3

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× b c d 1 ×

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3 Components

1 a 2 b 3 c 1 d 4 b 3 c 5 e 6 f 7 g

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Order within the scaffolding

For two 1-generated subcontexts Ks and Kt the relations Ist := I ∩ (Gs × Mt) and Its := I ∩ (Gt × Ms) are bonds. (A, B) ∈ B(Ks) and (C, D) ∈ B(Kt) follows (AII, AI) ≤ (C II, C I) ⇐ ⇒ C Its ⊆ B

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Diagram of the Scaffolding

1 2 e 4 abe f 3 c 5 e 6 a 7 g bd

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Thank you for your attention!

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