SMARANDACHE GROUPOIDS

Transcription

1 SMARANDACHE GROUPOIDS W. B. Vsnth Kndsmy Deprtment f Mthemtics Indin Institute f Technlgy Mdrs Chenni Indi. E-mil: vsntk@md.vsnl.net.in Astrct: In this pper we study the cncept f Smrndche Grupids sugrupids idel f grupids semi-nrml sugrupids Smrndche-Bl grupids nd Strng Bl grupids nd tin mny interesting results ut them. Keywrds: Smrndche grupid Smrndche sugrupid Smrndche idel f Smrndche grupid Smrndche semi-nrml grupid Smrndche nrml grupid Smrndche semi cnjugte sugrupids Smrndche Bl grupid Smrndche Mufng grupid Definitin []: A grupid (G ) is nn-empty set clsed with respect t n pertin (in generl need nt t e sscitive).. Definitin : A Smrndche Grupid G is grupid which hs prper suset S G which is semigrup under the pertin f G. Exmple : Let (G ) e grupid n mdul 6 integers. G = { 5} is given y the fllwing tle: Clerly S = { } S = { } nd S = { 5} re semigrups f G. S (G ) is Smrndche grupid. Exmple : Let G = {56789} e the set f integers mdul. Define n pertin n G y chsing pir ( 5) such tht = + 5 (md ) fr ll G.

2 The grupid is given y the fllwing tle Clerly S = { 5} S = { 6} S = { 7} S = { 8}nd S 5 = { 9} re semigrups under the pertin. Thus {G ( 5)} is Smrndche grupid. Therem : Let Z p = {... p -}. Define n Z p fr Z p y = + p (md p). {Z p (p)} is Smrndche grupid. Prf: Under the pertin defined n Z p we see S = { p} S = { p+} S = { p + }... S p = {p - p - } re semigrups under the pertin. Hence {Z p (p)} is Smrndche grupid. Exmple : Tke Z 6 = { 5}. ( 5) = (m n). Fr Z 6 define = m + n (md 6). The grupid is given y the fllwing tle: Every singletn is n idemptent semigrup f Z 6. Therem : Let Z p = {... p-}. Define n Z p y = + (p-) (md p) fr Z p. Then {Z p ( p - )} is Smrndche grupid. Prf: Under the pertin defined n Z p we see tht every element f Z p is idemptent therefre every element frms singletn semigrup. Hence the clim. Exmple : Cnsider Z 6 = {Z 6 ( 5)} given y the fllwing tle:

3 {} is semigrup. Hence is Smrndche grupid. It is esily verified tht Z 6 is Smrndche grupid s {Z 6 ( 5)} hs n idemptent semigrup {} under. Therem : Let Z p = {... p-} e the set f integers mdul p. Define n Z p y (p - ) + (p - ) (md p). Then {Z p (p - p - )} is Smrndche grupid. Prf: Z p = {... p - }. Tke (p - p - ) = frm Z p. Fr Z p define = (p - ) + (p - ) (md p). Clerly fr = = p we hve (p- )p + (p - )p = p(md p). Hence {p} is n idemptent semigrup f Z p. S {Z p (p - p - )} is Smrndche grupid. Definitin 5: Let (G ) e Smrndche grupid. A nn-empty suset H f G is sid t e Smrndche Sugrupid if H cntins prper suset K H such tht K is semigrup under the pertin. Therem 6: Nt every sugrupid f Smrndche grupid S is in generl Smrndche sugrupid f S. Prf: By n exmple. Let Z 6 = { 5} (md 6). Tke (t u) = ( 5) =. Fr Z 6 define n Z 6 y = t + u (md 6) given y the fllwing tle: Clerly {Z 6 ( 5)} is Smrndche grupid fr it cntins { } s semigrup. But this grupid hs the fllwing sugrupids:

4 A = { } nd A = { 5}. A hs n nn-trivil semigrup ({} is trivil semigrup). But A hs nn-trivil semigrup viz. {}. Hence the clim. Therem 7: If grupid cntins Smrndche sugrupid then the grupid is Smrndche grupid. Prf: Let G e grupid nd H G e Smrndche sugrupid tht is H cntins prper suset P H such tht P is semigrup. S P G nd P is semigrup. Hence G is Smrndche grupid. Definitin 8: i) A Smrndche Left Idel A f the Smrndche Grupid G stisfies the fllwing cnditins:. A is Smrndche sugrupid. Fr ll x G nd A x A. ii) Similrly ne defines Smrndche Right Idel. iii) If A is th Smrndche right nd left idels then A is Smrndche Idel. We tke {} s trivil Smrndche idel. Exmple 5: Let {Z 6 ( 5)} e Smrndche grupid. A = { 5} is Smrndche sugrupid nd A is Smrndche left idel nd nt Smrndche right idel. Esy t verify. Therem 9: Let G e grupid. An idel f G in generl is nt Smrndche idel f G even if G is Smrndche grupid. Prf: By n exmple. Cnsider the grupid G = {Z 6 ( )} given y the fllwing tle. 5 5 Clerly G is Smrndche grupid fr { } is semigrup f G. Nw {} is n idel f G ut is nt Smrndche idel s {} is nt Smrndche sugrupid. Definitin : Let G e Smrndche grupid nd V e Smrndche sugrupid f G. We sy V is Smrndche semi-nrml sugrupid if:. V = X fr ll G.. V = Y fr ll G. where either X r Y is Smrndche sugrupid f G ut X nd Y re th sugrupids.

5 Exmple 6: Cnsider the grupid G = {Z 6 ( 5)} given y the tle Clerly G is Smrndche grupid s {} is semigrup. Tke A = { 5}. A is ls Smrndche sugrupid. Nw A = A is Smrndche grupid. A = { }. { } is nt Smrndche sugrupid f G. Hence A is Smrndche semi-nrml sugrupid. Definitin : Let A e Smrndche grupid nd V e Smrndche sugrupid. V is sid t e Smrndche nrml sugrupid if V = X nd V = Y where th X nd Y re Smrndche sugrupids f G. Therem : Every Smrndche nrml sugrupid is Smrndche seminrml sugrupid nd nt cnversely. Prf: By the definitins nd we see every Smrndche nrml sugrupid is Smrndche semi-nrml sugrupid. We prve the cnverse y n exmple. In Exmple 6 we see A is Smrndche semi-nrml sugrupid ut nt nrml sugrupid s A = { } is nly sugrupid nd nt Smrndche sugrupid. Exmple 7: Let G = {Z 8 ( 6)} e grupid given y the fllwing tle: Clerly G is Smrndche grupid fr { } is semigrup f G. A = { 6} is Smrndche sugrupid. Clerly A = A nd A = A fr ll G. S A is Smrndche nrml sugrupid f G. 5

6 Definitin : Let G e Smrndche grupid H nd P e sugrupids f G we sy H nd P re Smrndche semi-cnjugte sugrupids f G if:. H nd P re Smrndche sugrupids. H = xp r Px r P = xh r Hx fr sme x G. Definitin : Let G e Smrndche grupid. H nd P e sugrupids f G. We sy H nd P re Smrndche cnjugte sugrupids f G if:. H nd P re Smrndche sugrupids. H = xp r Px fr sme x G.. P = xh r Hx fr sme x G. Exmple 8: Cnsider the grupid G = {Z ( )} which is given y the fllwing tle: Clerly G is Smrndche grupid fr { 6} is semigrup f G. Let A = {69} nd A = { 5 8 } e tw sugrupids. Clerly A nd A re Smrndche sugrups f G s { 6} nd { 8} re semigrups f A nd A respectively. Nw: A = { 5 8 } = A = { 6 9} nd similrly: A = { 6 9} = A. Hence A nd A re cnjugte Smrndche sugrupids f G. Definitin 5: Let G G G... G n e n grupids. We sy G = G G... G n is Smrndche direct prduct f grupids if G hs prper suset H f G which is semigrup under the pertins f G. It is imprtnt t nte tht ech G i need nt e Smrndche grupid fr in tht cse G will e viusly Smrndche grupid. Hence we tke ny set f n grupids nd find the direct prduct. 6

7 7 Definitin 6: Let (G ) nd (G' ) e ny tw Smrndche grupids. A mp φ frm (G ) t (G' ) is sid t e Smrndche grupid hmmrphism if φ ( ) = φ() φ() fr ll A. We sy the Smrndche grupid hmmrphism is n ismrphism if φ is n ismrphism. Definitin 7: Let G e Smrndche grupid. We sy G is Smrndche cmmuttive grupid if there is prper suset A f G which is cmmuttive semigrup under the pertins f G. Definitin 8: Let G e Smrndche grupid. We sy G is Smrndche inner cmmuttive grupid if every semigrup cntined in every Smrndche sugrupid f G is cmmuttive. Therem 9: Every Smrndche inner cmmuttive grupid G is Smrndche cmmuttive grupid nd nt cnversely. Prf: By the very definitins 8 nd 9 we see if G is Smrndche inner cmmuttive grupid then G is Smrndche cmmuttive grupid. T prve the cnverse we prve it y n exmple. Let Z = { } e integers mdul. Cnsider set f ll mtrices with entries frm Z = ( ) dente it y M. M =. M is mde int grupid y fr A = nd = in M. A B = = (md ) (md ) (md ) (md ) Clerly (M ) is Smrndche grupid fr =. S is semigrup.

8 8 Nw cnsider A = is Smrndche sugrupid ut A is nn-cmmuttive Smrndche grupid fr A cntins nn-cmmuttive semigrup S. S = such tht = nd =. S (M ) is Smrndche cmmuttive grupid ut nt Smrndche inner cmmuttive grupid. Definitin : A grupid G is sid t e Mufng grupid if fr every x y z in G we hve (xy)(zx) = (x(yz))x. Definitin : A Smrndche grupid (G ) is sid t e Smrndche- Mufng grupid if there exists H G such tht H is Smrndche sugrupid stisfying the Mufng identity: (xy)(zx) = (x(yz)x) fr ll x y z in H. Definitin : Let S e Smrndche grupid. If every Smrndche sugrupid H f S stisfies the Mufng identity fr ll x y z in H then S is Smrndche Strng Mufng grupid. Therem : Every Smrndche Strng Mufng grupid is Smrndche Mufng grupid nd nt cnversely. Prf: Every Strng Smrndche Mufng grupid is Smrndche Mufng grupid. The prf f the cnverse cn e prved y cnstructing exmples. Definitin : A grupid G is sid t e Bl grupid if ((xy)z)y = x((yz))y fr ll x y z G. Definitin 5: Let G e grupid. G is sid t e Smrndche-Bl grupid if G hs sugrupid H f G such tht H is Smrndche sugrupid nd stisfies the identity ((xy)z)y = x((yz)y) fr ll x y z in H. Definitin 6: Let G e grupid. We sy G is Smrndche Strng Bl grupid if every Smrndche sugrupid f G is Bl grupid. Therem 7: Every Smrndche Strng Bl grupid is Smrndche Bl grupid nd the cnverse is nt true. Prf: Ovius.

9 Therem 8: Let Z n = {... n-} e the set f integers mdul n. Let G = {Z n (t u)} e Smrndche grupid. G is Smrndche Bl grupid if t = t (md n) nd u = u (md n). Prf: Esy t verify. Exmple 9: Let G = {Z 6 ()} defined y the fllwing tle: { } is Smrndche sugrupid nd since = (md 6) nd = (md 6) we see G is Smrndche Bl grupid. Prlem : Let Z n = {... n-} e the ring f integers mdul n. G = {Z n (tu)} e grupid. Find cnditins n n t nd u s tht G:. is Smrndche grupid.. hs Smrndche semi-nrml sugrupids.. hs Smrndche nrml sugrupids.. is Smrndche cmmuttive. 5. is Smrndche inner cmmuttive. 6. is Smrndche-Bl grupid. 7. is Smrndche Strng Bl grupid. 8. is Smrndche-Mufng grupid. 9. is Smrndche-Strng-Mufng grupid.. hs lwys pir f Smrndche cnjugte sugrupids. References: [] R. H. Bruck A Survey f Binry Systems Springer Verlg 958. [] Rul Pdill Smrndche Algeric Structures Bulletin f Pure nd Applied Sciences Delhi Vl. 7 E. N ; [] W. B. Vsnth Kndsmy On rdered grupids nd its grupid rings J. f Mthemtics nd Cmp. Sci. Vl