V N U Joum al o f Science, M athem atics - P hysics 23 (2007) 221-224 Some problem on the shadow of segments inĩmite boolean rings T n H uye n, L e C ao Tu* D ep a rtm en t o f M a th em a tic s a n d C om puter Sciences U niversitỵ o f P edagogy, H o ch im in h City 745/2A L a c L ng Q uan, w a rd 10, D ìst Tan Binh, H o ch im in h City, Vietnam R eceived 18 S eptem ber 2007; received in revised form O ctober 2007 A b s t r a c t In this p a p e r , vve consider finite B oolean rings in vvhich w ere defm ed tw o orders: natural o rder and an tilexicographic order T he m ain result is concem ed to the notion o f shadovv o f a segm ent We sh all prove som e necessary and su íĩic ie n t conditions for the shadow o f a segm ent to be a segm ent Introduction C onsider a íinite B oolean ring: B ( n ) = {x = XỊX2 -Xn : Xi € { , }} w ith natural o rder < /v defined by X rifc + 1, there m ust be a n u m b e r M such tha t nic + < m/c - = M Choose C o r o lla r y d = ( l, ? , k - l, M ) £ B (n ,k ), w e the re fo re have [ d ; b ] c [a ;b ] N o te that the segm ent [d ;b ] sa tisíys c o n d itio n s o f c o ro lla ry 2.2, w e n o w im ita te the above p ro o f to fin is h the c o ro lla ry C e rta in tly , the last c o ro lla ry is a s o lu tio n fo r o u r ke y questions, in the case (b ) W h a t about the re m a in in g case ? F irs t o f a ll, w e tu m o u r a tte n tio n to the case (a) and have tha t: Let a,b e B(n,k) be elements such that a = { ĩ i \ , n fc _ i, M ) andb = ( m i , TTik-1 , M ) then A [ a , ] is a segment if and only if m \ = M - k + 1and either Tik- < M — \ or nic- — k —2 Proof T ake c =a - M ; d = b - M e B ( n , k - l) then A [ a ; b} = [c; d] u { x + M : X e A [c , d]} Suppose tha t A [ a , ] is a segm ent then there m u st have g = ( l, ? , k - l) € A [ c ; ể ] and v?(cỉ) + l = with Uk-s+1 = M- S + then a = ( n i , n/c-s, M - s + | •••) M) and d = ( m i, M — s + , M ) Take as re q uired In th e case rik- 3+ = M - s + , w e consider fir s t s = Since À = { x + ( A / - s + , M ) : i e Á[a' + ( M - S + ) ; and h ’ = ( M - k + l, ,M - s ) e and o n ly ( m if the u n io n IS (d ) i , m k - 3) < \JA , where a ’= ( n i , U k- 3) It is clear that the Union IS (d ) u A [ a ; / ỉ + M Ị is a segment i f is a segm ent N o te that mfc - < M - s, therefore b' — h ' H e n c e , th e la st r e q u ir e m e n t is e q u iv a le n t to th e re q u ir e m e n t th a t ip ( a ') < i p ( b ') + l and A [ a ' + ( A / - S + ); the theorem B (n ,k -s ) h ' + ( M - s + 1)}} h' + ( M - s + 1)] = Ịa '; / i ' ] u { y + ( M - s + l ) : y € A [ a '; / i '] } is a segment B y , the la tte r is e q u iv a le n t to the requirem ents tha t nfc_s < M —s or rik- - = k - s ~ T h e p ro o f is co m p lete d References [1] I.Anderson,Combinatorics o f JInìte sets, Clarendon Press, Oxford, (1989) [2] B.Bolloba? Combinatorics, Cambridge University Press, (1986) [3] G o H.Katona, A ứìeorem on ílnite sets In Theorỵo/Graphs Proc Colloq Tihany, Akadmiai Kiado Academic Press, New York (1966) pp 187-207 [4] J B Kniskal, The number of simpliccs in a complex, In Mathematicaỉ optimization techniques (cđ R Bcllman ), ưnivcrsity of Calíomia Press, Bcrkcley (1963) pp 251-278 ... vin g an in d e x fro m the elem ent in A T h e conception about the shadow o f a set was used e flfìcie n tly by m any m a the m a ticia n s as: Spem er, K ru s k a l, Katona, C lem ent, ? We s... lity, vve fin is h the prove o f the lem m a A s an im m ed ia te consequence,we get the fo llo w in g Let a,b £ B(n,k) be elements such that a = (ỉ k-1, M) and b =(M-k+l then the shadow A [ a ,... I.Anderson,Combinatorics o f JInìte sets, Clarendon Press, Oxford, (1989) [2] B.Bolloba? Combinatorics, Cambridge University Press, (1986) [3] G o H.Katona, A ứìeorem on ílnite sets In Theorỵo/Graphs