The rooted circuits and the rooted cocircuits of convex geometries, closure operators, and monotone extensive operators
Kashiwabara Kenji and Nakamura Masataka
Department of System Sciences, University of Tokyo, Komaba, Meguro, Tokyo 153‐8902, JAPAN.
1 Outline 1. Closure spaces, closure systems, matroids, and convex geometries 2. Supersolvable antimatroids 3. Lattice-embedding of convex geometries to convex geometries 4. Rooted circuits and implicational systems 5. Extensive and intensive operators 6. Relations of areas to convex geometries
2 1. Closure spaces, closure systems, matroids, and convex geometries
Let E be a non‐empty finite set.
Def. Closure operator :2E 2E A (A) (extensive) A B (A) (B)(monotone) ((A)) (A) (idempotent)
Def. (, E)is called a closure space.
3 Def. Closure system Def. Closure operator E E K 2E :2 2 ・EK, ・A(A) ・ ・A B (A) (B) X,Y K X Y K ・ ( (A)) (A) A member of a closure system is called a closed set.
(A) X K{AE: (A)A} XK,AX one-to-one correspondence
4 Ex. A closure system is a lattice under inclusion relation.
{1,2,3,4,5}
{1,2,4,5} {2,3,4,5} {1,2,3,5} {2,4,5} {1,2,5} {2,3,5}
{1,5} {3,5} {2,5}
{5}
5 Closure spaces (closure systems)
Def. For each element eE,suppose X Ee and X is minimal with respect to the property e (X). Then we say (X,e) is a rooted circuit of a closure operator , (or more generally a monotone extensive operator ). X is called the stem, and e is called the root. We denote the collection of the rooted circuits of by C().
Ex. Rooted circuits {x,y,z} ({y,z}, x) ({x,z}, {x} {y} {z} ({x,y},y) z) 6 Closure spaces (closure systems)
Conversely, the rooted-circuit system C ( ) determines the closure operator .
(A) A{eE A:(X,e)C(), X A} (A E).
If (A contains) an element eEAhere is a minimal A X Asuch that e(X) wirh (X,e) C. e X
7 Closure spaces (closure systems) Def. Rooted cocircuit For e E and Y E e , if Y is minimal with respect to the property e (E(Y e) ), (Y,e) is said to be a rooted cocircuit. We denote the collection of the rooted cocircuits by D().
Ex. {x,y,z} Rooted Cocircuits ({y}, x) ({z}, x) {x} {y} {z} ({z}, y) ({x}, y) ({x}, z) ({y}, z) a(E {y, x})({z}){z} 8 Ex. Affine-point configuration E {a,b,c,e} 2 R (A) conv.hull(A) E
Rooted Circuits a ({a,b,c},e) e b c Rooted Cocircuits ({a},e), ({b},e), ({c},e).
9 Closure spaces (closure systems)
Ex. Affine-point configuration E {a,b,c,d,e} R2 Rooted Circuits (A) conv.hull(A) E ({a,b,c},e), ({c,d},e), a ({a,b},d). d e Rooted Cocircuits ({c},e), ({a,d},e), b c ({b,d},e). ({a},d), ({b},d) 10 Closure spaces (closure systems) Def. (A) {eA:e(Ae)} (A E) Is the extreme-point operator of a closure operator .
Ex. Affine-point configuration E {a,b,c,d,e} R2.
a ({a,b,c,d,e}){a,b,c} d ({a,c,d,e}){a,c,d} e b c ({a,b,e}){a,b,e}
11 Matroid
Def. A closure space (,E) A is a matroid if b
a(Ab) b(Aa) (Exchange Property)
a
Abstraction of linear dependency
12 Matroid a(Ab) b(Aa) (Exchange Property)
Ex. A graphic matroid
b b
x x w y w a y a u z u z
A{x, y,z} (A) {x, y,z,u,w}
13 Ex. A graphic matroid M(G) of a graph G E=E(G) b Rooted circuits with the root z x ({a, b, x, u}, z), y w a ({x, w, u}, z), ({w, y}, z), u z ({a, b, y}, z).
Proposition In a matroid M, for any circuit C and any element eC, (Ce, e) is a rooted circuit, and vice versa.
14 Convex geometry
Def. A closure space (, E)is a convex geometry if the corresponding closure operator satisfies the anti‐exchange property. a b, a,b(A), b(Aa) a(Ab)
Ex. Affine‐point configuration E R2. (Anti‐exchange Property) (A) conv.hull (A) E
(A) b a Abstraction of convexity (Jamison 1982)
a(Ab) b(Aa) (Excange Property) 15 Convex geometry
Theorem ( Edelman and Jamison 1985)
Let (, E) be a closure space , K be the closure system associated with it, and be the extreme‐point operator. Then the following are equivalent. (1) is a closure operator with the anti‐exchange property.
(2) For any closed set X K with X E, there exists e E X such that X e K.
(3) ((A)) (A) for A E.
(4) ( (A)) (A) for A E. [ Krein‐Milman property] 16 Convex Geometries and Antimatroids
Def. Convex geometry Def. Antimatroid K 2E F 2E ,EK, ・ٛ F・ ・X,Y K X Y K ・A,BF ABF ・X K, X E ・AF, A eE X : X eK eA: AeF
Def. Antimatroid An element of K is called a F {E X : X K} convex set, and an element of F is a feasible set.
17 E
X e A X Ae
convex geometry antimatroid
18 {a,b.c,b} {a,b,c,d}
{b,c,d} {a,c,d} {a,b,d} {c,d} {a,b} {c} {a} {b} A convex An antimatroid geometry
19 classes of convex geometries ・ Affine convex geometry ・ poset convex geometry ・ double shelling of a poset ・ simplicial shelling of a chordal graph ・ tree node‐shelling convex geometry ・ graph search convex geometry generalized affine convex geometries = all the convex geometries 20 Principle of shelling process of antimatroids Once an element is removable, it remains removable until it is deleted.
(A) is the set of removable elements in A.
A : E ; while( (A) is non-empty) do delete any element in (A) from A end
21 Ex. Poset double shelling antimatroid – Repeating the deletion of a minimal or maximal element. c {a,b,c,d} d {a,b,c,d} a b {a,c,d} {a,c} {a,b,d} {a,b,c} {a,c,d} {b,c,d} c {a} {a,b} {a,c} {b,c} {c,d} d a b {a} {b} {c} A ｐoset double shelling antimatroid
22 Ex. poset convex geometries
c {a,b,c,d} {a,b.c,b} d {a,b,d} {b,c,d} {a,c,d} a b {c,d} {a,b} Hasse diagram of a poset P {a} {b} {c} Deleting a minimal antimatroid convex geometry element of P of P antimatroid
23 Def. Monophonically convex sets of a chordal graph G : a graph, V(G): the vertex set of G X V(G) is monophonically convex if it holds that a,b X and c V is on a chordless path between a and b c X
Theorem (Edelman and Jamison 1985) The collection of monophonically convex sets is a convex geometry if and only if G is a chordal graph.
24 Ex. Monophonically convex sets of a chordal graph
125 {12345}
{2345} {1345} 3 4 {245} {345} {135} A chordal graph G {24}{25} {45} {34} {35} {15} {13}
{2} {4} {5} {3} {1} Removable elements =Simplicial vertex
A convex geometry of the chordal graph G
25 Antimatroids and rooted cocircuits
Def. Let e E . In an antimatroid F , suppose that an element H F contains e and is minimal with respect to this property. Then we say that ( Y , e ) is a rooted cocircuit where Y H e . e is called the root and Y is the costem.
[Remark] H is necessarily a join-irreducible element in the lattice F .
26 Ex. Poset double shelling antimatroid – Repeating the deletion of a minimal or maximal element. {a,b,c,d} c {a,b,d} {a,b,c} {a,c,d} {b,c,d} {a,b} {a,c} {b,c} {c,d} d a b {a} {b} {c} antimatroid
Element: minimal feasible set Rooted cocircuits a {a} ( , a), b {b} ( , b), c {c} ( , c) d {a, b, d}, {c, d}, ({a, b}, d), ({c}, d)
27 clutter
Def. A clutter is a collection of sets in which no member contains another properly.
Def. For a clutter L on E, a subset of E is a transversal of L if it intersects every member of L . The collection of the minimal transversals called the blocker of L and denoted by b ( L ). [Note] If a set intersects every member of a clutter L , then it contains a member of the blocker b(L). 28 clutter
Ex. clutters and their blockers L {{a, c},{b, c}}, b (L) {{c},{a, b}}, b(b(L)) L
L {{a, b},{b, c},{c,a}}, b (L) {{a,b},{b,c},{c,a}}, b(b(L)) L
Proposition For any clutter L, b (b (L)) L holds.
29 Rooted circuits: root stems costems ({a,c}, d), ({b,c}, d) a ー b ー Rooted cocircuits: c ー ( , a), ( , b), ( , c) d {a,c}, {b,c} {a,b}, {c} ({a,b}, d), ({c}, d)
L={ {a, c}, {b, c} }, b(L)={ {c}, {a, b} }
Proposition (Korte, Lovasz and Schrader) For a convex geometry (K,E) , let C be the collection of the rooted circuits, and D be the collection of the rooted cocircuits. Then for each eE, C(e) {X :(X,e)C}and D(e) {Y :(Y ,e)D}are the blocker of each other. 30 Def. Meet‐distributive lattice: A finite lattice is meet‐distributive if for any xL, the interval [x, x1 x2 xk ] is a Boolean lattice where x1, x2, , xk are the elements covered by x.
L x Theorem (Edelman 1986) x x x A finite lattice is meet-distributive 1 2 k if and only if it is isomorphic to the lattice of a convex geometry. x1 x2 xk
31 Theorem (Dilworth, Boulay, Edelman, et al.) Let L be a finite lattice. Then the following are equivalent. (1) Every element of L has a unique irredundant join decomposition to join‐irreducible elements. (2)L is lower semimodular, and every modular sublattice is distributive. (3) L is meet distributive. (4) L is lower semimodular and join‐semidistributive. (5)L is isomorphic to a closure lattice of a convex geometry.
32 2. Supersolvable antimatroids Def. F 2E is an antimatroid if and only if K {E A: AF} is a convex geometry.
Def. F 2E is an antimatroid Def. F 2Eis an antimatroid if if (1)F , (1)F , (2) A,BF ABF, (3) If AF and A , (2A) If A,BF and A B , then eA such that then bB\ A such that AeF . AbF . [strong axioms of antimatroids]
33 Def. A lattice is supersolvable if there is a maximal chain D (called an M-chain) such that a sublattice generated by D and any other chain is necessarily distributive. Def. An antimatroid is supersolvable if it is supersolvable as a lattice.
Ex. Supersolvable and not supersolvable lattices
M-chain A supersolvable lattice A non-supersolvable lattice
34 Def. For a convex geometry, we define a digraph, called the circuit-graph, with the vertex set being the underlying set and the edge set {( f , e ) : ( X , e ) C , f X } where C is the family of the rooted circuits.
Ex. a bc Circuit-graph b
Rooted circuits a c ( {a, c}, b) Acyclic digraph
35 Ex. An affine point configuration 3 4 5 Circuit graph 5 2 1 1 5 4 Rooted circuits 2 4 123 ({1,2,3}, 5) , ({1,4}, 5) (acyclic partial order) ({2,3}, 4) 3
36 Ex. The circuit graph is not bcd a acyclic in this case. b Rooted circuits ( {a, c}, b), ( {a, d}, b) a d ( {a, d}, c), ( {b, d}, c) c Theorem For an antimatroid F 2E, the following are equivalent. (1) F is supersolvable. (2) There exists a total ordering on E such that
for A,BF with A B and bmin (B\ A) , AbF. (Armstrong 2009) (cf. (2A)) (3) The rooted‐circuit digraph of the convex geometry
K FC {E A: AF}is acyclic.
Suppose that the circuit digraph of K FC {E A: AF} is acyclic. Then it determines a partial order P on E. Then Corollary Any linear extension of P satisfies (2) above. 37 3. Lattice‐embedding of convex geometries to convex geometries
Lemma
Let K1 , K2 be convex geometries on E, and 1, 2 be their closure operators. Then the following are equivalent.
(1)K1 K2 .
(2)2(A) 1(A) for A E .
(3) For any rooted circuit (X2,a) of K2 , there is a rooted
circuit (X1,a) of K1 such that X2 X1 .
38 Theorem E K K Let K1 , K2 2 be convex geometries on E such that 1 2.
Then the maximum size of stems of K1 is at most the maximum size of stems of K2 .
Conjecture Let (K1,E1), (K2,E2) be convex geometries.
Suppose there exists a lattice‐embedding f : K1 K2. Then the maximum size of stems of K1 is at most the
maximum size of stems of K2 .
39 A result from the conjecture
Let (K1,E1), (K2,E2 ) be convex geometries. If the maximum size of stems of K2 is larger than that of K1 , then there exists no lattice‐embedding f : K1 K2 .
Theorem (Adaricheva et al. (2003)) A finite join‐semidistributive lattice cannot be necessarily embedded into a finite biatomic atomistic convex geometry.
40 Def. A lattice with zero is biatomic if ・ every element is above an atom (atomic), ・ for every atom p of L and all element a,bL, if p a b , then there are atoms q a and r b such that p qr. We misunderstood that the size of a stem of a biatomic convex geometry is at most two. Ex. 7 points configuration in an affine space below. Deleting a rooted circuit ( {a,b}, e), then the other rooted circuits defines a convex geometry which is biatomic and has a stem of size 3. a
e
b 41 5. Extensive and intensive operators Def. An operator is a map f :2E 2E Def. An operator f is extensive if A f (A). Def. An operator f is intensive if f (A) A. extensive f f *(A ) {aA:a f (Aa)} intensive
* intensive f f (A) A {bE A:b f (Ab)} extensive
Ex. * closure (A ) (A) extreme‐point operator {a A : a (A a)} operator
42 Def. Ext(E) : the collection of all the extensive operators. Def. Int(E) : the collection of all the intensive operators.
Theorem (Danilov and Koshevoy 2009) Ext(E) * Int(E) is a bijection.
Def. An extensive operator is monotone if B A(B) (A)
43 Def. An intensive operator is hereditary if B A (A) B (B)
Ex. Selection of representatives World
(A) is the national football team of A (Hungary) (A) (B)
(B) is the representative B A team of football of B (Szeged)
44
`hereditary’ is a nice property as a choice function. Def. Mon(E) : the collection of all the monotone extensive operators on E
Def. Her(E) : the collection of all the hereditary intensive operators on E Ext(E) * Int(E) U U Mon(E) * Her(E)
Theorem (Danilov and Koshevoy 2009) Mon(E) * Her(E) is a bijection.
45 Ext(E)
Mon(E)
Anti‐exchange idempotent = Exchange property Closure operator property
Anti‐exchange Exchange closure operator Closure operator
Convex geometry Matroid
46 Extensive operators
Monotone extensive operator
Closure operator
Convex Matroid geometry
47 Def. An intensive operator is path‐independent if
(AB) ((A) (B)) for any A, B E. A B (ABD) ((A) (B) (D)) West East JAPAN
Theorem (Koshevoy 1999) An intensive operator (choice function) is path‐independent if and only if it is an extreme‐point operator of a convex geometry.
48 Def. A rooted set on E is a pair (X,e) such that eE, X Ee.
Def. A rooted clutter C on E is a family of rooted sets such that for each eE, C(e) { X :(X,e)C} is a clutter.
Def. A pair (C,D) of rooted clutters is a rooted circuit‐cocircuit system if for each eE, C(e) {X :(X,e)C } and D(e) {Y :(Y ,e)D} are the blocker of each other. We say that C and D are point‐ wise blocker of each other.
Conversely, a monotone extensive operator determines a rooted circuit-cocircuit system.
49 Def. For a rooted circuit‐cocircuit system (C,D), let (A) Inc (A) A {eE A:(X,e)C, X A} (A E), C Tran (A) A {eE A:(Y,e)D,Y A} (A E). D
point‐wise blocker D C A Tran Inc C D e X Mon (E) e (X) for (X,e) C
Theorem The diagram is commutative. 50 and (A) tran (A) {eA:(X,e)C, X A} C inc (A) {eA: (Y,e)D, Y A} D
point‐wise blocker Theorem C D This diagram is commutative. tran inc C D
Her (E)
51 Proposition For a rooted circuit‐cocircuit system, Inc Tran tran inc C D Mon(E),C D Her(E), * , and * .
pointwise pointwise blocker C blocker D Circ(E) Cocir(E)
Inc Tran tran inc Inc Tran tran inc * Mon(E) * Her(E)
Theorem The diagram is commutative, and every arrow is a bijection.
52 Monotone extensive operator
Closure operator
Convex matroid geometry Exchange property Anti‐exchange property
53 Anti‐exchange monotone extensive operator
Theorem Let (C,D) be a circuit‐cocircuit system, and Inc Tran Mon(E). C D Then the following are equivalent.
(AEx) satisfies the anti‐exchange property. (CA) If (X,x)C and (Y, y)C, then either (X',x)C for
some X' (X Y) y or (Y', y)C for some Y' (X Y)x. (DA) If (X,x)D and (Y, y)D, then either (X',x)D for
some X' (X Y) y or (Y', y)D for some Y' (X Y)x. 54 Note: Switching C and D gives nothing changed. Closure operator
Theorem Let (C,D) be a circuit‐cocircuit system, and Inc Tran C D Mon(E). Then the following are equivalent. (Clo) is idempotent, i.e. is a closure operator. (CO) If (X,x)C, (Y, y)C and yX, then there exists
(W, y)C such that W (X Y)x. (DO) If (X,x)C and f X , then there exists (Y, f )D
such that Y (X f ) x.
55 Exchange monotone extensive operator
Theorem Let (C,D) be a circuit‐cocircuit system, and Inc Tran Mon(E). C D Then the following are equivalent.
(Ex) satisfies the exchange property. (CE) If (X,x)C and f X, then there exists (Y, f )C
such that Y (X f ) x. (DE) If (X,x)D, (Y, y)D and yX, then there exists
(W, y)D such that W (X Y)x.
56 For a circuit‐cocircuit system (C,D), we shall call (D, C) the dual system. Under taking the dual, the anti‐exchange property is invariant. Taking the dual switches the exchange property and the idempotent (closure operator). Hence the concept of matroid is invariant under taking the dual. When (C,D) gives rise to a matroid, the dual system(D, C) gives the dual matroid in the ordinary sense.
57 Exchange property + closure = matroid Exchange Property
(CE) If (X,x)C and f X, then there exists (Y, f )C
such that Y' (X f ) e. (DE) If (X,x)D, (Y, y)D and yX, then there exists
(W, y)D such that W (X Y)x. Closure
(CO) If (X,x)C, (Y, y)C and yX, then there exists
(W, y)C such that W (X Y)x. (DO) If (X,x)C and f X , then there exists (Y, f )D
such that Y' (X f ) e.
58 New self‐dual axioms of Matroids Axioms of Matroid (CE) If (X,x)C and f X , then there exists (Y', f )C
such that Y' (X f ) x. (DO) If (Y, y)D and f Y, then there exists (Y', y)D
such that Y'(Y f ) x.
Axioms of Matroid
(CO) If (X, x)C, (Y, y)C and y X, then there exists
(W, y)C such that W (X Y) x. (DE) If (X, x)D, (Y, y)D and y X, then there exists
(W, y)D such that W (X Y) x. 59 6. Related areas of convex geometries
Closure lattices (Lattice theory) = closure systems
Greedoid Meet‐distributive (Combinatorial Geometric lattices lattices optimization) = Matroid: Convex geometries Anti‐ Matroid: closed sets
(Abstraction of convexity) matroid independent
sets Path‐independentU choice function (Choice function theory) Choice functions 60 4. Rooted circuits and implicational systems
Def. An implicational system
S {(Ai ,Bi ) : iI } (Ai ,Bi E) We write A B S instead of (A, B) S.
Def. X E fulfills if AX BX.
Lemma K(S) {X E: X fulfills everyABS } is a closure system.
61 Def. For a closure system K if K K ( S ) then S is a generating system. If K(S) K(S'), then we say that S and S ' are equivalent. A minimal generating system is called a basis.
Def. A basis S of K is minimum if | S | | S ' | for any basis S ' of K .
Def. A basis S of K is optimal if (| A|| B|) (| A'|| B'|) for any basis S ' of K . ABS A'B'S'
62 Def. Let C be the collection of rooted circuits of K We put Stem(K) { X : (X,e)C}
Lemma Let K be a closure system, and C be the collection of rooted circuits. Then S(C) { X e : (X,e)C } is a generating system of K.
63 Def. quasiclosed set: For AE, A A {(X) : X A, (X) (A)}, A A A A . AE is quasiclosed if A A and A (A).
Def. pseudoclosed set: P E is a pseudoclosed set if P is a minimal quasiclosed set among those quasiclosed sets Q satisfying (P) (Q).
64 Theorem (Guigues and Duquenne 1986, Wild 1994) Let ( , E ) be a closure system.
(1) The basis S P { P ( ( P ) P ) : P is pseudoclos ed} is a minimum basis of ( , E) (1) An optimal basis is always a minimum basis. (2) For each pseudoclosed set P , every optimal basis
includes an implication X P Y P with XP P.
The cardinality of X P is uniquely determined as
| XP |min{| X | : X P, (X) (P)}.
65 Def. For a stem X , we put Int (X) { f : (X, f )C}, bd (X) (X)(X int(X)). If bd (X) , then X is called a prime stem.
Ex. Affine point configuration.
a Stems
e {a,b,c} not prime d {a,c,d} prime {a,d) prime b c
66 Theorem Suppose that the sizes of stems of a closure space are all the same. Then X is a stem if and only if it is a pseudoclosed set.
Theorem Under the same condition as above. Then S {X ((X) X ): X is a stem} is a minimum basis.
67 Theorem For an affine convex geometry ( ,E), S {X ((X) X ): X is a prime stem} is a minimum basis.
Theorem For an affine convex geometry ( ,E),
S{ X eX : X is a prime stem} is an optimal basis.
68 Thank you. Koeszenem szepen.
69 What is study? The Master said, `Is it not a pleasure to learn and practice what is learned? Is it not a joy to have friends come together from far? Is it not gentlemanly not to take offence though men may take no note of him?’ ‐‐‐ The Analects of Confucius
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