Independence Posets Nathan Williams (UTD) Joint with Hugh Thomas - PowerPoint PPT Presentation

Independence Posets Nathan Williams (UTD) Joint with Hugh Thomas (UQAM) Triangle Lectures in Combinatorics November 14, 2020

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Breakout Paid Only Video Antitrust Rooms MS Teams ✓ ✓ ✓ Zoom ✓ ✓ Google Meet ✓ ✓ US Postal Service ✓ General: a set of objects J and properties M , with maps � � m : J → 2 M an object �− → its properties � � j : M → 2 J a property �− → objects with that property

Rudolf Wille’s Formal Concept Analysis subsets of objects �− → shared properties subsets of properties �− → common objects

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∈ 2 J × 2 M such that: � � General: a concept is a pair X , Y � � � m X = m ( x ) = Y x ∈ X � � � j Y = j ( y ) = X . y ∈ Y

The poset of concepts is the set of all concepts ordered by � � � X ′ , Y ′ � iff X ⊆ X ′ . X , Y ≤ a b c d 3 1 2 3 4 a b c a ✓ ✓ ✓ b ✓ ✓ 2 3 1 3 4 c ✓ ✓ a c a b d d ✓ 2 3 4 a 1 2 3 4

The poset of concepts is the set of concepts ordered by � � � X ′ , Y ′ � ≤ X , Y iff X ⊆ X ′ or (equivalently) Y ′ ⊆ Y . a b c d 3 a b c 2 3 3 4 1 a b a c d 2 3 4 a 1 2 3 4 Theorem (Lattice Representation I) The poset of concepts is a lattice , and every lattice arises as a poset of concepts.

Lattices “Never in the history of mathematics has a mathemat- ical theory been the object of such vociferous vitupera- tion as lattice theory. Dedekind, Jónsson, Kurosh, Malcev, Ore, von Neumann, Tarski, and most prominently Garrett Birkhoff have contributed a new vision of mathematics, a vision that has been cursed by a conjunction of misunder- standings, resentment, and raw prejudice.” —Gian-Carlo Rota ( The Many Lives of Lattice Theory )

Definition A ( finite ) poset L is a lattice if each pair x , y ∈ L have: a join (least upper bound): x ∨ y , and a meet (greatest lower bound): x ∧ y . “Like its elder sister group theory, lattice theory is a fruitful source of abstract concepts, common to traditionally unrelated branches of mathematics. Both subjects are based on postulates of an extremely simple and general nature. It was this which convinced me from the first that lattice theory was destined to play—indeed, already did play implicitly—a fundamental role in mathematics. Though its importance will probably never equal that of group theory, I do believe that it will achieve a compa- rable status.” —Garrett Birkhoff ( Lattice Theory , Second Edition).

Definition ◮ An element j ∈ L is join-irreducible if when j = � S , then j ∈ S . Write J for the set of all join-irreducibles. ◮ An element m ∈ L is meet-irreducible if when m = � S , then m ∈ S . Write M for the set of all meet-irreducibles. • • • • • • •

For x ∈ L , write J ( x ) = { j ∈ J : j ≤ x } M ( x ) = { m ∈ M : m ≥ x } . • a b c d 3 3 a b c 2 1 2 3 1 4 3 4 c a c b d a b d 2 3 4 a a • 1 2 3 4

Since any element is the join of the join-irrs below it and the meet of the meet-irrs above it: Theorem (Lattice Representation II) Every finite lattice L is isomorphic to (both) ◮ { J ( x ) : x ∈ L } under inclusion, and ◮ { M ( x ) : x ∈ L } under reverse inclusion.

1 2 3 4 a b c d • 3 2 1 4 c b d Show j ≤ m a •

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Historical Notes ◮ Formal concept analysis was Rudolf Wille’s “restructuring” of lattice theory (published 1982). ◮ Previously appeared in the 1973 thesis of George Markowsky as the poset of irreducibles . ◮ First (?) appeared in 1965 work of Marc Barbut as l’algèbre des techniques d’analyse hiérarchique . ◮ Similar ideas due to M. P. Schützenberger.

George Markowsky’s Extremal Lattices “When do lattices have a ‘compact’ representation theorem?”

• 1 2 3 4 a ✓ ✓ ✓ 3 b ✓ ✓ c ✓ ✓ 2 1 4 d ✓ c b d a • Definition (Markowsky) � � � � � J � M A lattice is extremal if � = � = length L . � � � �

• 1 2 3 4 3 a ✓ ✓ ✓ b ✓ ✓ 2 1 4 c ✓ ✓ c b d d ✓ a • Key point: A chain of maximal length ˆ 0 = x 0 ⋖ x 1 ⋖ · · · ⋖ x n = ˆ 1 lets us pair up J and M by j i ↔ m i , where: x i = j 1 ∨ · · · ∨ j i = m i +1 ∧ · · · ∧ m n

A chain of maximal length lets us pair up J and M : a ↔ 1 , b ↔ 4 , c ↔ 2 , d ↔ 3 d 3 1 4 2 3 1 2 3 4 a ✗ ✓ ✓ ✓ �→ �→ b ✗ ✓ ✓ ✗ b 4 c 2 c ✗ ✗ ✓ ✓ d ✓ ✗ ✗ ✗ a b c d a 1

Theorem (Markowsky) Every finite extremal lattice L is isomorphic to the lattice of maximal orthogonal pairs of a finite directed acyclic graph G.   X ∩ Y = ∅    � �  X , Y : no arrow X → Y    X , Y maximal  d 3 ordered by b 4 c 2 a 1 iff X ⊆ X ′ or � � � X ′ , Y ′ � ≤ X , Y Y ′ ⊆ Y

Theorem (Markowsky) A finite graded extremal lattice L is distributive, in which case G is the comparability graph of a poset. Corollary: Birkhoff’s theorem for distributive lattices . 1 2 3 4 a ✓ ✓ ✓ b ✓ ✓ c ✓ ✓ d ✓

Theorem (Markowsky) A finite graded extremal lattice L is distributive, in which case G is the comparability graph of a poset. Corollary: Birkhoff’s theorem for distributive lattices . 1 2 3 4 a ✓ ✓ ✓ b ✓ ✓ c ✓ ✓ d ✗

Theorem (Markowsky) A finite graded extremal lattice L is distributive, in which case G is the comparability graph of a poset.

Hugh Thomas’s Trim Lattices “What if a distributive lattice weren’t graded?”

Call a relation ( X , Y ) ≤ ( X ′ , Y ′ ) in an extremal lattice overlapping if Y ∩ X ′ � = ∅ . 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 Theorem (Thomas-W.) An extremal lattice is trim iff every relation is overlapping iff every cover relation is overlapping.

Call a relation ( X , Y ) ≤ ( X ′ , Y ′ ) in an extremal lattice overlapping if Y ∩ X ′ � = ∅ . Theorem (Thomas-W.) An extremal lattice is trim iff every relation is overlapping iff every cover relation is overlapping. This lets us label every cover relation.

D : x �→ { g ∈ G : y ⋖ x } Theorem (Thomas-W.) U : x �→ { g ∈ G : x ⋖ y } are each bijections from the elements of a trim lattice L to the independent sets of G.

Independence Posets “What if a distributive lattice weren’t a lattice?’ Click for interactive version

Recall: For trim lattices L , ◮ Every cover relation is overlapping ( Y ∩ X ′ � = ∅ ) and ◮ There are bijections D : x �→ { g ∈ G : y ⋖ x } U : x �→ { g ∈ G : x ⋖ y } from L to the independent sets of G . Questions: ◮ How do D and U fit together? ◮ Can we simply express cover relations on D and U ?

Fix G a finite directed acyclic graph. Definition (Thomas-W.) A pair ( D , U ) of independent sets of G is orthogonal if D ∩ U = ∅ and there is no arrow D → U . An orthogonal pair is tight if whenever ◮ an element of D is increased 1 2 3 4 ◮ an element of U is decreased 1 2 3 4 ◮ an element is added to D 1 2 3 4 ◮ an element is added to U 1 2 3 4 the result isn’t orthogonal.

Theorem (Thomas-W.) Any independent set I can be uniquely completed to a top ( I , U ) and a top ( D , I ) .

Definition (“Flips”) If g ∈ D or g ∈ U , flip g ( D , U ) is defined by: (i) fix all elements of D not < g (ii) fix all elements of U not > g (iii) swap g from D to U or vice-versa (iv) complete D and U (uniquely). g g Flip g �− − − →

Theorem (Thomas-W.) The independence poset of G is the poset with cover relations ( D , U ) ⋖ flip g ( D , U ) . 6 5 3 4 2 1 6 1 3 6 6 5 5 3 4 3 4 2 2 1 1 5 1 6 6 6 6 5 5 5 3 4 3 4 3 4 2 2 2 1 1 1 6 3 4 6 6 5 5 2 3 5 3 4 3 4 2 2 1 1 4 3 1 6 6 6 5 5 5 3 4 3 4 3 4 2 2 2 1 1 1 1 6 2 6 6 5 5 3 4 3 4 2 2 1 1 3 6 1 6 5 3 4 2 1

Theorem (Thomas-W.) If an independence poset is a lattice, it is a trim lattice. If it is further a graded lattice, it is a distributive lattice. “What if a distributive lattice weren’t a lattice?”

Thank You!