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

The Scaffolding of a Formal Context Stephan Doerfel 1 , 2 1 Knowledge and Data Engineering Group, Department of Electrical Engineering and Computer Science, University of Kassel 2 Department 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 Motivation 1 The Scaffolding of a Lattice 2 The Scaffolding of a Formal Context 3 Example and Diagram 4 Stephan Doerfel (University Kassel) Scaffolding CLA 2010 2 / 31

Agenda Motivation 1 The Scaffolding of a Lattice 2 The Scaffolding of a Formal Context 3 Example and Diagram 4 Stephan Doerfel (University Kassel) Scaffolding CLA 2010 3 / 31

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 Stephan Doerfel (University Kassel) Scaffolding CLA 2010 4 / 31

Example - Lattice Stephan Doerfel (University Kassel) Scaffolding CLA 2010 5 / 31

Example - Scaffolding Stephan Doerfel (University Kassel) Scaffolding CLA 2010 6 / 31

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? Stephan Doerfel (University Kassel) Scaffolding CLA 2010 7 / 31

Agenda Motivation 1 The Scaffolding of a Lattice 2 The Scaffolding of a Formal Context 3 Example and Diagram 4 Stephan Doerfel (University Kassel) Scaffolding CLA 2010 8 / 31

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 Stephan Doerfel (University Kassel) Scaffolding CLA 2010 9 / 31

Construction means 1/2 Residual Map For complete lattices L 1 and L 2 and a surjective inf-morphism α : L 1 → L 2 , the map x �→ inf α − 1 x α : L 2 → L 1 , is an injective sup-morphism, called the residual map to α . Separating Maps A set of maps α t : L → L t ( t ∈ T ) is called separating if for all x � = y ∈ L there is an index t such that α t ( x ) � = α t ( 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 sup S such that sup S A = s ⇐ ⇒ sup A = s holds for A ⊆ S and s ∈ S . Theorem For an arbitrary index set T, complete lattices L and L t ( t ∈ T ) and separating complete homomorphisms α t : L → L t S ( α t | t ∈ T ) := { α t α t x | 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 ( α t | t ∈ T ) . Stephan Doerfel (University Kassel) Scaffolding CLA 2010 11 / 31

Scaffolding 2/2 Scaffolding The scaffolding of a complete lattice L of finite length is S ( L ) := S ( α t | t ∈ T ) = { α t α t x | x ∈ L , t ∈ T } \ { 0 } where L t are all completely subdirectly irreducible factors of L α t are all surjective complete homomorphisms α t : L → L t 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 αα x = x . Stephan Doerfel (University Kassel) Scaffolding CLA 2010 12 / 31

Agenda Motivation 1 The Scaffolding of a Lattice 2 The Scaffolding of a Formal Context 3 Example and Diagram 4 Stephan Doerfel (University Kassel) Scaffolding CLA 2010 13 / 31

Means Now, given a formal context, we need equivalents for separating morphisms residuals subdirectly irreducible factors subirreducible elements Stephan Doerfel (University Kassel) Scaffolding CLA 2010 14 / 31

Bonds K x = ( G x , M x , I x ) Bond A bond from K s to K t is a relation R st ⊆ G s × M t , such that g R st is an intent of K t for every object g ∈ G s and m R st is an extent of K s for every attribute m ∈ M t . Stephan Doerfel (University Kassel) Scaffolding CLA 2010 15 / 31

Bonds and morphisms α : ( A , B ) �→ ( A R st I t , A R st ) is a sup-morphism B ( K s ) → B ( K t ) M s M t α : ( A , B ) �→ ( B R st , B R st I s ) is a inf-morphism B ( K t ) → B ( K s ) K s G s R st α is residual to α α is a complete homomorphism iff a bond R ts exists such that K t G t R ts ∀ ( A , B ) ∈ B ( K s ) holds A R st I t = B R ts . Such a pair ( R st , R ts ) we will call hom-bonds . Stephan Doerfel (University Kassel) Scaffolding CLA 2010 16 / 31

Products of bonds Product and Composition For bonds R rs from K r to K s and R st from K s to K t the bond product is defined by R rs ◦ R st := { ( g , m ) ∈ G r × M t | g R rs I s ⊆ m R st } For the corresponding sup-morphisms holds: φ R rs ◦ R st = φ R st ◦ φ R rs . Definition A set of hom-bonds ( R t , S t ) between K und K t is called separating if for any two extents A � = C of K there is an index t ∈ T such that A R t � = C R t holds. Stephan Doerfel (University Kassel) Scaffolding CLA 2010 17 / 31

Construction supremum-dense subset For separating hom-bonds ( R t , S t ) between K and K t ( t ∈ T ) S ( R t , S t ) t ∈ T := { ( A R t ◦ S t I , A R t ◦ S t ) | ( A , B ) ∈ B ( K ) , t ∈ T } \ { ( M I , M ) } is a supremum-dense subset of B ( K ) . Theorem For contexts K and K t , separating hom-bonds ( R t , S t ) between K and K t and their corresponding homomorphisms α t : B ( K ) → B ( K t ) holds S ( α t | t ∈ T ) = S ( R t , S t ) t ∈ T . Stephan Doerfel (University Kassel) Scaffolding CLA 2010 18 / 31

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 ⇒ ( ∀ h ∈ G : g I ⊆ h I , g I � = h I = g ւ m : ⇐ ⇒ ( h , m ) ∈ I ) , ⇒ ( ∀ n ∈ M : m I ⊆ n I , m I � = n I = g ր m : ⇐ ⇒ ( g , n ) ∈ I ) , g ր ւ m : ⇐ ⇒ g ւ m and g ր m . Stephan Doerfel (University Kassel) Scaffolding CLA 2010 19 / 31

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 � g � K = ( G g , M g , I g ) containing g , called the 1 -generated subcontext of g in ( G , M , I ). Stephan Doerfel (University Kassel) Scaffolding CLA 2010 20 / 31

The Scaffolding Separating hom-bonds If K is a reduced context of a doubly founded concept lattice and � g � K = ( G g , M g , I g ), then ( R g , S g ) with R g := I ∩ ( G × M g ) and S g := I ∩ ( G g × M ) ( g ∈ G ) are separating hom-bonds. Definition If B ( K ) is the doubly founded concept lattice of a reduced context K and ( R g , S g ) are as above, then the relative sup-semilattice S ( R g , S g ) g ∈ G = { ( A R g ◦ S g I , A R g ◦ S g ) | ( A , B ) ∈ B ( K ) , g ∈ G } \ { ( M I , M ) } is called the scaffolding of K and will be denoted by S ( K ). Stephan Doerfel (University Kassel) Scaffolding CLA 2010 21 / 31

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 ) . Stephan Doerfel (University Kassel) Scaffolding CLA 2010 22 / 31

Simplification S ( R g , S g ) g ∈ G = { ( A R g ◦ S g I , A R g ◦ S g ) | ( A , B ) ∈ B ( K ) , g ∈ G } \ { ( M I , M ) } Simplified Scaffolding For a reduced context K of a doubly founded concept lattice with 1-generated subcontexts K t ( t ∈ T ) covering K holds { ( C II , C I ) | ( C , C I t ) ∈ B ( K t ) , C I t � = M t } . � S ( K ) = t ∈ T The sets { ( C II , C I ) | ( C , C I t ) ∈ B ( K t ) , C I t � = M t } ( t ∈ T ) are called the components of the scaffolding. Stephan Doerfel (University Kassel) Scaffolding CLA 2010 23 / 31

Agenda Motivation 1 The Scaffolding of a Lattice 2 The Scaffolding of a Formal Context 3 Example and Diagram 4 Stephan Doerfel (University Kassel) Scaffolding CLA 2010 24 / 31

f b d a e g 3 7 4 6 2 5 1 Stephan Doerfel (University Kassel) Scaffolding CLA 2010 25 / 31