Pure Mtemtl Senes, Vol. 4, 205, no., 29-36 HIKARI Ltd, www.m-kr.om ttp://dx.do.org/0.2988/pms.205.4028 Loue re Mg Squres over Mult Set o Rtonl Numers s n Innte Feld A. M. Byo Deprtment o Mtemts nd Computer Sene Fulty o Sene, Federl Unversty Ksere, Gome Stte, Nger G. U. Gr Deprtment o Mtemts Fulty o Sene, Amdu Bello Unversty, Zr, Kdun Stte, Nger Copyrgt 204 A. M. Byo nd G. U. Gr. Ts s n open ess rtle dstruted under te Cretve Commons Attruton Lense, w permts unrestrted use, dstruton, nd reproduton n ny medum, provded te orgnl work s properly ted Astrt Ts work s poneerng nvestgton o te Loue re Mg Squres over mult set o rtonl numers o te orm { : 0, Z} nnte lger eld, were Z denotes te mult set o nteger numers. By te Loue re Mg Squres L (s we denote t), we understnd te set o mg squres ormed y te De L Loue re Proedure. It s explted tt te set equpped wt te mtrx nry operton o ddton (s we denote t) orms n nnte ddtve eln group, nd te set enlosed wt te rtonl numers multplton (s we denote t) orms n nnte multpltve eln group te underlnng set so onsdered o te entres o te orementoned squres s te mult set o te orementoned set o numers. (L,, ) orms n nnte eld. Mtemts Suet Clsston: 2 xx Keywords: Loue re Mg Squres, Mult Set, Innte Feld, Addtve Aeln Group, Multpltve Aeln Group
30 A. M. Byo nd G. U. Gr. Introduton Te Loue re Mg Squres L equpped wt te mtrx nry operton o ddton orms semgroup te underlnng set onsdered s te mult set o nturl numers. I we underlne te mult set o nteger numers s entres o te squre, (L, ) orms n nnte ddtve eln group. Te Loue re Mg Squres over te mult set o rtonl numers o te orm { : 0, Z} orms n nnte multpltve eln group, (L, ), tus, mkng (L,, ) n nnte eld. Ts s not te explton o te denton o te eld presented n [], ut te two re nlogous. Artrrly, 5 5 Sem Pndgonl Loue re Mg Squres s onsdered not squres su tt n 7 to eonomy spe, nd not 3 3 su squre to dodge ner s oe. Te mult set o rtonl numers o te orm{ : 0, Z} s ptly onsdered even toug te mult set o rtonl numers o te orm { : 0, s xed n Z nd Z }, spel type o slr multplton, wll do. Also, we onsder te sequene,,, n tmes,,,, n tmes,,,, n tmes, rter tn,,,... n tmes,,,,... n tmes,... toug te lter wll lso gve n nlogous result; ut presentng ot te two s ys tutology. 2. Prelmnres Denton 2.. A mult set s set n w repetton o elements s relevnt. For exmple, {,,,,,,,, } {,, } {,,,,,,,, } or ter Loue re Mg Squres re not somorp. Denton 2.2. A s mg squre o order n n e dened s n rrngement o rtmet sequene o ommon derene o rom to n 2 n n n n squre grd o ells su tt every row, olumn nd dgonl dd up to te sme numer, lled te mg sum M(S) expressed s M(S) = n3 +n nd entre pee C s C = M(S). 2 n Denton 2.3. Mn Row or Column s te olumn or row o te Loue re Mg Squres ontnng te rst term nd te lst term o te rtmet sequene n te squre.
Loue re mg squres over mult set o rtonl numers 3 2.4 Loue re Proedure (NE-W-S or NW-E-S, te rdnl ponts) Consder n empty n n squre o grds o ells. Strt, rom te entrl olumn or row t poston n were s te greter nteger numer less tn or equl 2 to, wt te numer. Te undmentl movement or llng te squres s dgonlly up, rgt (lok wse or NE or SE) or up let (nt lok wse or NW or SW) nd one step t tme. I lled ell (grd) s enountered, ten te next onseutve numer moves vertlly down wrd one squre nsted. Contnue n ts son untl wen move would leve te squre, t moves due N or E or W or S (dependng on te poston o te rst term o te sequene) to te lst row or rst row or rst olumn or lst olumn. See lso [2] or su proedure. Te squre o grd o ells [ ] n n s sd to e Loue re Mg Squre te ollowng ondtons re stsed. n = n =. = k T. tre[ ] = tre[ n n ] = k n n. n,, n 2 2, n, n, n 2 2 re on te sme mn olumn or row nd n,n, n 2 2 n, n re on te sme mn olumn or row,, 2 2 were s te greter nteger less or equl to, T s te trnspose (o te squre), k s te mg sum (mg produt s dened nlogously) usully expressed s k = n [2 + (n )] rom te sum o rtmet sequene, were s te 2 ommon derene long te mn olumn or row nd s te rst term o te sequene nd n 2 n 2 = k. n 2.5 Group A non empty set G togeter wt n operton * s known s group te ollowng propertes re stsed.. G s losed wt respet to *..e., G,, G.. s ssotve n G..e., ( ) = ( ),,, G.. e G, su tt e = e =, G. Here e s lled te dentty element n G wt respet to *. v. G, G su tt = = e, were e s te dentty element. Here s lled te nverse o nd smlrly vse vers. Te nverse o te element s denoted s.
32 A. M. Byo nd G. U. Gr Te ove denton o group s gven n []. I n ddton to te ove xoms, te ollowng xom s stsed, we ll (G, ) n eln group were (G, ) s denotton o group. v. =,, G. Tt s ll (not some o) te elements o G ommutes. 2.6 Feld I (G, ) s n ddtve eln group nd (G, ) s multpltve eln group, ten (G,, ) s eld. 3. Te Loue re Mg Squres Feld Te underle mult set o te nnte ddtve Loue re Mg Squres eln group (L, ) s {: Z} nd te underle mult set o te nnte multpltve Loue re Mg Squres eln group (L, ) s { : 0, Z}. Te wole onept s sed on te mnestton o n n Loue re Mg Squres were n s odd vng ot te mg sum nd te mg produt. {,,, n tmes,,,, n tmes,.. or,, n tmes,,,, n tmes... :,, Z} s te mult set onsdered, were n = 2Z + + nd Z + s te set o postve ntegers (greter tn or equl to ) or n = s trvlty nd n = 2, te oddest prme, does not exst. Te orrespondng set o mg squres o entres te sequene o elements o te mult set ove re L n were n = 3, 5, 7,... L 3 [ ] or { [ ] :,, Z },
Loue re mg squres over mult set o rtonl numers 33 L 5 { d e e d d e d e [ d e ] or d e [ e d d e d e d :,, Z e ] } L 7, L 9, L, re onstruted nlogously. Teorem 3.. (L n, ) s n ddtve eln group. Proo. Artrrly onsderng L 5, we dene s ollows: Let A, B L 5 were d e e d A = d e d e [ d e ] x y u v w y u v w x nd B = u v w x y v w x y u [ w x y u v] d e e d A B = d e d e [ d e ] = C x y u v w y u v w x u v w x y v w x y u [ w x y u v] d + x e + y + u + v + w e + y + u + v + w d + x = + u + v + w d + x e + y + v + w d + x e + y + u [ + w d + x e + y + u + v]. L 5 s losed wt respet to : From te ove denton, A, B L 5, ten A B = C L 5. Ts s more vvd y lettng (sy) p = + u, q = + v, r = + w, s = d + x nd t = e + y.. s ssotve: For A, B, C L 5, A (B C) = (A B) C wene + ( + ( + )) = ( + ) + ( + ) were A, B nd + C (rtrrly oosen. )
34 A. M. Byo nd G. U. Gr. 0 0 0 0 0 0 0 0 0 0 Te ddtve dentty element I s 0 0 0 0 0 0 0 0 0 0 were e 0 exept te [ 0 0 0 0 0] rst entry o te squre s seres o 0s orrespondngly n tmes. v. E element o L 5 s n nverse: I D L 5, ten D L 5 were D s te squre wt mg sum M(S) ormed s result o slr multplton o entres n D vng mg sum M(S) y. For exmple, te nverse o [ ] s [ ]. v. s ommuttve: A, B L 5 A B = B A wene te mtrx nry operton o ddton over set o nteger numers s ommuttve (nertne). Tus, (L n, ) s n nnte ddtve eln group. Teorem 3.2. (L n, ) s n nnte multpltve eln group. l m k m k l Proo. Let D, E L 5. We dene s ollows: Let D = k l m k l m [ k l m ] nd E = e g e g e g. Ten D E = l m e kg m e kg l e kg l m = F g e [ g e ] kg l m e [ kg l m e ]. L 5 s losed wt respet to : For D, E L 5, F L 5 rom te ove denton. Ts s vvd y ntmte look t te pttern o elements n F: e,, kg, l, nd m.. Assotvty: s ssotve or A, B, C L 5 A (B C) = (A B) C euse x(y(xy)) = (xy)(xy) were x A, y B nd xy C. Ts s x, y nd xy.
Loue re mg squres over mult set o rtonl numers 35. Te dentty element I s were e [ ] I exept te rst entry o te squre s produt o s orrespondngly n tmes. g g v. E element o L 5 s n nverse: I G = g L 5, ten H = g [ g] [ g g g g g] su tt G H = H G = I v. s ommuttve: A, B L 5 A B = B A sne te entres n te squre re rtonl numers nd rtonl numers multplton s ommuttve, we re done. Tus, (L n, ) s n nnte multpltve eln group. Teorem 3.3. (L n,, ) s n nnte eld. Proo. Ts ollows mmedtely rom Teorem 3. nd Teorem 3.2 ove. 4. Conluson Every Loue re Mg Squre under dsuss n ts work s out 4 msellny eets o rottons nd/or reletons. Consderng out o te 4 eets s rtrry. Altoug te Loue re Mg Squres presented ere re not te s (ovous) ones, yet tey re sem pndgonl Loue re wt unque mg sums nd produts. We reommend tt you re pt n serng or n exmple o ny lger strutures e t semgroup (Fon), group (Symmetr), eld (Loue re ),vetor spe(mg Squres n generl) or not enter Loue re Mg Squres, you wll nd one.
36 A. M. Byo nd G. U. Gr Reerenes [] Sreernn K. S, V. Mdukr Mllyy, Sem Mg Squres s Feld, Interntonl Journl o Alger, 6 (202), 249-256. [2] Gn Yee Sng, Fon Wn Heng, Nor Hnz Srmn, Propertes nd Solutons o Mg Squres, Menemu Mtemtk (Dsoverng Mtemts), 34 (202), 69. Reeved: Novemer 8, 204; Pulsed: Deemer 2, 204