Decomposition of Cellular Balleans

Decomp osition of Cellular Balleans I. V. Pr otasov, A . Tsvietkova Abstract. A ballean is a set endo w ed with some famil y of its subsets which a re called the balls. W e p ostulate the prop erties of the family of balls in such a w a y that the balleans can b e considered as the asymptotic counterparts o f the uniform top ologi cal spaces. The isomorphisms in the category of balleans are called asy- morphisms. Every metric space can b e considered as a ballean. The ultrametric spaces are p rotot ypes for th e cellular balleans. W e prov e some general theorem abou t d ecomposition of a homogeneous cel lular ballean in a direct pro duct of a p oin ted family of sets. App lying t his th eorem we show that the balleans of t w o uncountable gro ups of the sa me regular cardinality are asymorphic. A b al l structur e is a triple B = ( X, P , B ) where X, P are non-empt y sets, and for all x ∈ X and α ∈ P , B ( x, α ) is a subset of X wh ic h is calle d a b al l of r adius α aroun d x . It is sup p osed that x ∈ B ( x, α ) for all x ∈ X , α ∈ P . The set X is called the supp ort of B , P is called the se t of r ad ii . Giv en any x ∈ X , A ⊆ X , α ∈ P , w e p ut B ∗ ( x, α ) = { y ∈ X : x ∈ B ( y , α ) } , B ( A, α ) = [ a ∈ A B ( a, α ) , (1) B ∗ ( A, α ) = [ a ∈ A B ∗ ( a, α ) . (2) A ball structur e B = ( X , P , B ) is called a b al le an (or a a c o arse structur e ) if • ∀ α, β ∈ P ∃ α ′ , β ′ ∈ P suc h that ∀ x ∈ X B ( x, α ) ⊆ B ∗ ( x, α ′ ) , B ∗ ( x, β ) ⊆ B ( x, β ′ ); • ∀ α, β ∈ P ∃ γ ∈ P suc h that ∀ x ∈ X B ( B ( x, α ) , β ) ⊆ B ( x, γ ); Let B 1 = ( X 1 , P 1 , B 1 ) and B 2 = ( X 2 , P 2 , B 2 ) b e balleans. 1 A mapping f : X 1 → X 2 is called a ≺ -mapping if ∀ α ∈ P 1 ∃ β ∈ P 2 suc h that: f ( B 1 ( x, α )) ⊆ B 2 ( f ( x ) , β ) . A bijectio n f : X 1 → X 2 is called an asymorphism b et w een B 1 and B 2 if f and f − 1 are ≺ -map p ings. In th is case B 1 and B 2 are called asymorphic . If X 1 = X 2 and the ident it y mapping id: X 1 → X 2 is an asymorphism, w e identify B 1 and B 2 and wr ite B 1 = B 2 . F or motiv ation to study balleans, see [1], [2], [3], [4]. Ev ery metric space ( X , d ) d etermines the metric b al le an B ( X, d ) = ( X, R + , B d ), wh ere R + is the set of non-negativ e r eal n um b ers, B d ( x, r ) = { y ∈ X : d ( x, y ) ≤ r } . A ballean B is called metrizable if B is asymorp hic to B ( X , d ) for some metric ballean. By [3,Theorem 2.1], a ballean B is metrizable if and only if B is connected and the cofinalit y cf( B ) ≤ ℵ 0 . A ballean B = ( X , P , B ) is c onne cte d if, for an y x, y ∈ X , there exists α ∈ P such that y ∈ B ( x, α ). T o defin e cf( B ), we use the natural preordering on P : α ≤ β if and only if B ( x, α ) ⊆ B ( x, β ) for ev ery x ∈ X . A subset P ′ is c ofinal in P if , for ev ery α ∈ P , there exists α ′ ∈ P ′ suc h that α ≤ α ′ , so cf( B ) is the minimal cardinalit y of cofinal su bsets of P . Giv en an arbitrary ballean B = ( X, P , B ), x, y ∈ X and α ∈ P , we say that x, y are α -p ath c onne cte d if th ere exists a finite s equ ence x 0 , x 1 , ..., x n , x 0 = x , x n = y su c h that x i +1 ∈ B ( x i , α ), for ev ery i ∈ { 0 , 1 , ..., n − 1 } . F or an y x ∈ X and α ∈ P , we put B ✷ ( x, α ) = { y ∈ X : x, y are α -path conn ecte d } The ballean B ✷ = ( X, P , B ✷ ) is called the c el lularization of B . A b allea n B is called c el lular if B ✷ = B . F or c h aracte rizations of cellular balleans see [3, Chapter 3]. Example 1 . A metric d on a set X is called an ultr ametric if d ( x, y ) ≤ max { d ( x, z ) , d ( y , z ) } for all x, y , z ∈ X . If (X,d) is an ultrametric space then the balle an B ( X , d ) is cellular. Moreo v er, b y [3, Theorem 3.1], a b allea n B is metrizable and cellular if and only if B is asymorphic to th e metric ballea n B ( X, d ) of some ultrametric sp ace ( X, d ). Example 2 . Le t G be an infinite group w ith the identit y e , κ b e an infinite card in al such that κ ≤ | G | , F ( G, κ ) = { A ⊆ G : e ∈ A, | A | < κ } . 2 Giv en any g ∈ G and A ∈ F ( G, κ ), we put B ( g , A ) = g A and get the ballean B ( G, κ ) = ( G, F ( G, κ ) , B ). In the case κ = | G | , w e write B ( G ) instead of B ( G, κ ). A ballea n B ( G, κ ) is cellular if and only if either κ > ℵ 0 or κ = ℵ 0 and G is locally finite (i.e. ev ery finite subset of G is con tained in some finite sub group). Example 3 . A family of subsets of a group G is called a Bo ole an gr oup ide al if • A, B ∈ ℑ ⇒ A ∪ B ∈ ℑ ; • A ∈ ℑ , A ′ ⊂ A ⇒ A ′ ∈ ℑ ; • A, B ∈ ℑ ⇒ AB ∈ ℑ , A − 1 ∈ ℑ ; • F ∈ ℑ for ev ery finite subset F of G . Ev ery Boolean group ideal ℑ on G determines the ballean B ( G, ℑ ) = ( G, ℑ , B ), where B ( g , A ) = g A f or all g ∈ G, A ∈ ℑ . The balleans on groups determined b y the Bo olean group ideals ca n b e considered (see [3 , Ch ap- ter 6 ]) as the asymptotic co unt erparts of the group topologies. A balle an B ( G, ℑ ) is cellular if and only if ℑ h as a base consisting of th e sub groups of G . A connected ballean B = ( X, P , B ) is cal led ord inal if there exists a cofinal w ell-ordered (by ≤ ) subset of P . Clea rly , ev ery metrizable ballean is ordinal. Theorem 1 . L et B = ( X, P , B ) b e an or dinal b al le an. Then B is either metrizable or c el lular. Pr o of . If cf( B ) ≤ ℵ 0 then B is metrizable b y theorem 2.1 from [3]. Assume that cf( B ) > ℵ 0 . Giv en an arb itrary α ∈ P , w e c ho ose indu ctiv ely a sequence ( α n ) n ∈ ω in P suc h that α 0 = α and B ( B ( x, α n ) , α ) ⊆ B ( x, α n +1 ) for ev ery x ∈ X . Since cf( B ) > ℵ 0 , w e can pic k β ∈ P s u c h that β ≥ α n for ev ery n ∈ ω . Then B ✷ ( x, α ) ⊆ B ( x, β ) for ev ery x ∈ X , so B ✷ = B . Let γ b e an ordinal, { Z λ : λ < γ } b e a family of non-empt y sets. F or ev ery λ < γ w e fix some element e λ ∈ Z λ and sa y that the family { ( Z λ , e λ ) : λ < γ } is p ointe d . A dir e ct pr o duct Z = ⊗ λ<γ ( Z λ , e λ ) is the set of all functions f : { λ : λ < γ } → ∪ λ<γ Z λ suc h that f ( λ ) ∈ Z λ and f ( λ ) = e λ for all but finitely many λ < γ . W e consider the b all structure B ( Z ) = ( Z , { λ : λ < γ } , B ), where B ( f , λ ) = { g ∈ Z : f ( λ ′ ) = g ( λ ′ ) for all λ ′ ≥ λ } It is easy to ve rify that B ( Z ) is a cellular b allea n. W e sa y that a ballean B i s de c omp osable in a dir e ct pr o duct if B is asymorphic to B ( Z ) for some direct pro duct Z . 3 Theorem 2 . L et γ b e a limit or dinal, B = ( Z, { λ : λ < γ } , B ) b e a b al le an such that: ( i ) B ✷ ( x, α ) = B ( x, α ) for al l x ∈ X , α ∈ P ; ( ii ) if α < β < γ then B ( x, α ) ⊂ B ( x, β ) for e ach x ∈ X ; ( iii ) if β is a limit or dinal and β < γ then B ( x, β ) = ∪ α<β B ( x, α ) for e ach x ∈ X ; ( iv ) ther e exists a c ar dinal κ 0 such that B ( x, 0) = κ 0 for e ach x ∈ X ; ( v ) for every α < γ ther e exists a c ar dinal κ α such that eve ry b al l of r adius α + 1 is a disjoint union of κ α -many b al ls of r adius α . Then B is de c omp osable in a dir e ct pr o duct. Pr o of . W e fix some set Z 0 of cardinalit y κ 0 and define in d uctiv ely a family of sets { Z α , α < γ } . If α is a limit ordinal, w e tak e Z α to b e a singleton. If α = β + 1 we tak e a set Z α of cardinalit y κ β . F or ev ery α < γ , w e c ho ose some elemen t e α ∈ Z α , put Z = ⊗ λ<γ ( Z λ , e λ ) and show that B is asymorphic to B ( Z ). T o th is end we fix some elemen t x 0 ∈ X and, f or ev ery α < γ , define a mapp ing f α : B ( x 0 , α ) → ⊗ β ≤ α ( Z β , e β ) suc h that, for all β < α < γ , f α | B ( x 0 ,β ) = f β and th e ind u ctiv e limit f of the family { f α : α < γ } is an asymorphism b et w een B and B ( Z ). Here we id en tify ⊗ β ≤ α ( Z β , e β ) with the corresp onding subset of ⊗ β <γ ( Z β , e β ). A t the fir st step w e fix some bijection f 0 : B ( x 0 , 0) → Z 0 suc h that f 0 ( x 0 ) = e 0 . Le t us assu me that, for some α < γ , w e ha v e defin ed the mappings { f β : β < α } . I f α is a limit ordinal, w e p ut f α : B ( x 0 , α ) → ⊗ β <α ( Z β , e β ) to b e an in d uctiv e limit of the family { f β : β < α } . Since Z α = { e α } we can iden tify ⊗ β <α ( Z β , e β ) with ⊗ β ≤ α ( Z β , e β ), so f α : B ( x 0 , α ) → ⊗ β ≤ α ( Z β , e β ). If α = β + 1, by cellularit y of B , there exists a su b set Y ⊆ B ( x 0 , α ) , x 0 ∈ Y su c h that B ( x 0 , α ) is a disjoint union of the fam- ily { B ( y , β ) : y ∈ Y } . F or ev ery y ∈ Y , we ca n rep eat the ind uctiv e pro cedure of construction of f α : B ( x 0 , β ) → ⊗ λ ≤ β ( Z λ , e λ ) to define a map- ping f ′ β ,y B ( y , β ) → ⊗ λ ≤ β ( Z λ , e λ ). Thus w e fix some b ijection h : Y → Z α , h ( x 0 ) = e α and put f β ,y ( x ) = ( f ′ β ,y ( x ) , h ( y )) , x ∈ B ( y , β ). A t last, giv en any x ∈ B ( x 0 , α ), w e c ho ose y ∈ Y suc h that x ∈ B ( y , β ) and put f α ( x ) = f β ,y ( x ). By the constru ctio n of f as an in ductiv e limit of the f amily { f α : α < γ } , giv en any x ∈ X an d α < γ , we hav e f ( B ( x, α )) = B ( f ( x ) , α ) so f is an asymorph ism. In the n ext tw o coroll aries and Th eorem 3 B ( G ) is a b allean defined in Example 2. 4 Corollary 1. L et G b e a c ountable lo c al ly finite gr oup. Then B ( G ) is de c omp osable in a dir e ct pr o duct of finite sets. Pr o of . W e write G as a union G = ∪ n<ω G n of a n increasing c hain of finite groups. Clearly , B ( G ) is asymorphic to the ballean B = ( G, ω , B ) where B ( g , n ) = gG n . W e put κ 0 = | G 0 | , κ n +1 = | G n +1 : G n | a nd apply Theorem 2. Corollary 2. L e t G b e an unc ountable gr oup of r e gular c ar dinality γ . Then B ( G ) is de c omp osable in a dir e ct pr o duct. Pr o of . W e write G as a union G = ∪ α<γ G α of an in cr easing c hain of subgroups su c h that | G 0 | = ℵ 0 , | G α | < γ and G α = ∪ β <α G β for ev ery limit ordinal α . Since γ is regular, ev ery subset F ⊂ G , | F | < | G | is conta ined in some subgroup G α . It follo ws that B ( G ) is asymorphic to the ballean B = ( G, γ , B ), where B ( g , α ) = g G α . Apply Theorem 2. Theorem 3 . L et G, H b e two unc ountable gr oups of the same r e gular c ar dinality γ . Then B ( G ) and B ( H ) ar e asymor phic. Pr o of . W e consider t wo cases. Case 1: γ is a limit cardinal. W e c h o ose an increasing family { G α : α < γ } of subgroups of G su c h that G = ∪ α<γ G α , | G 0 | = ℵ 0 , | G α +1 | = | G α | + and G β = ∪ α<β G α for ev ery limit ordinal β . Put κ α = | G α | + , α < γ . By Theorem 2, B ( G ) is asymorph ic to B ( Z ) w h ere the dir ect pro du ct Z is defined b y the family of cardinals { κ α : α < γ } . Since H admits a filtration H = ∪ α<γ H α with the same family { κ α : α < γ } of parameters, B ( H ) is also asymorphic to B ( Z ). Case 2: γ = λ + for some cardinal λ . W e write G as a union G = ∪ α<γ G α of an increasing family of sub groups su c h that | G α | = λ , | G α +1 : G α | = λ for every α < γ , and G β = ∪ α<β G α for every limit ordinal β . Pu t κ α = λ for ev ery α < γ . By Th eorem 2, B ( G ) is asymorphic to B ( Z ), where Z is defi ned by the family of parameters { κ α : α < γ } . Since H admits a filtration with the same family of parameters, B ( H ) is also asymorp hic to B ( Z ). This completes the p ro of. It should b e m en tioned that Theorem 3 do es not hold for coun table groups. By [2, Theorem 10.6], there exists a f amily F of coun table lo cally finite groups su c h that an y t wo groups from F are non-asymorphic and | F | = 2 ℵ 0 . W e d o not kno w if Coroll ary 2 and T heorem 3 are true for groups of singular cardinalities. 5 References [1] A. N. Dranishniko v, A sympto tic top olo g y , Rus s ian Mat h Surveys 55 (2000 ), no. 6, 1085-112 9. [2] Protaso v I., Banakh T. Bal l Str uctur es and Colorings of Gr aphs and Gr oups , Matematic al Studies Monograph Series, 11 . L’viv: VNTL Publishers, 2003. [3] Ihor Protaso v and Mic hael Z aric hnyi, Gener al Asymptolo gy , Mathemat- ical S tu dies Monograph Series, 12. L ’viv: VNTL Pub lishers, 2007. [4] John Ro e, L e ctur es on Co arse Ge ometry , Un iv ersit y Lecture Series, 31. Pro vid ence, RI: AMS, 2003. I.V. Protaso v ( pr otas ov @ unicy b.k iev.ua ) Departmen t of Cyb ernetics, Kyiv National Univ ersit y , V olo dimirsk a 64, Kiev 01033, UKRAINE Anastasiia Tsvietk ov a ( ts v ietk ov a @ math.utk .edu ) Departmen t of Mathematics, Univ er s it y of T ennessee, 121 Ayres Hall, Kno xville, T ennessee 37996-1 300, US A 6