some elementary algebraic considerations inspired by the smarandache function

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  • 8/12/2019 SOME ELEMENTARY ALGEBRAIC CONSIDERATIONS INSPIRED BY THE SMARANDACHE FUNCTION

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    SO L ~ R Y ALGEBRAIC CONSIDERATIONSINSPIRED BY THE SXARANDACBE FONCTION

    byE.Radescu, N.Radescu, C.Dumitrescu

    I t i s known that the Smarandache function S: N - - ~ f i I I ,S{n) .min{kln d iv ides k } sa t i s f ies

    (i) S is surject ive( i i ) S([m,n]) max { S(m),s (n)} , where [m,n] is the smallest

    common multiple of m and n.That is on N there are considered both of the div is ib i l i ty

    order Sd ( mS d n i f and only i f m divide n ) and the usual order s.Of course the algebraic :.lsual operations "+" and n . play also animportant ro le in the descr ipt ion of the proper t ies of S.For instance i t is said that :1]:

    max { S (kll) , S nD) } :5 S ( kn) Iul) s nS kll) +kS nD) If we consider the universal algebra ~ , Q ) , with

    C-{Yd,clJO} where Yt1 : N*)2---N* is given by, m V. n-[m,n] and4ao : {N):)---N*, i s given by 4 a o { ~ } ) = l = e v < l and analogously the

    l;niversal algebra N*, Q ) with Q'=N, -o}, where V: (N*)2 __-N*, is

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    def ined by mVn=max{m,n}, and 0 : N*)o---N* i s def ined by

    0 ({.,}) =l=ey , then it r e su l t s :1.PROPOSITION. LetN={S- k) IkN*},whereS- k) ={xN*IS x) =k}. Then

    a) N i s countab le ca rdN*= a l e f ze ro ) .b) on N may be def ined an un ive rsa l a lgebra , i somorfe with

    N* , Of)Proof . b) Let i): N)2---N be def ined bY i) S- a) , S-(b =S-(c) ,

    where C=S(xVQY), with xS- a) , y S - b ) .Then w i s wel l def ined because i x1S-(a) Y1ES-(b) theS 1S.VdYl) =S(1S.)VS(Y1) = a V b = S ~ x ) V S y ) =S(xVQY) =c.

    Example. i ) S - 23 ) ,S- 14= S- 23 ) because i fo r ins tancex=46 S- 23) and y=49 S- 14) t hen 46 Vd49 =2254 and 8 2254) =23.

    In fac t , because c=S(xVQY) =S(x) VS(y) =aVb, it r e su l t s t ha tw i s def ined by

    i) S- a ) , 8- b ) =S-(aVb) .We de f ine now i)o: N) )---N by c . . > o { ~ } ) =8- 1 )

    Let us note 8- 1) =e ,,). ThenT/S-(k) EN i) S- k) , ef) =6) e ,,), S-(k =S-(k)

    Then N , 0 ) i s an un ive rsa l a lgebra i Q={CI),6)oLI t may be de f ined h N---N* an isomorphism between N, 0 and

    N* , a/ , by h S- (k) =k.

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    e haveVS- a) ,S- b ) EN h 6 l S- a) ,S- b =h S- aVb) ==aVb=h S- a Vh S- b) )

    that i s h is a morphism.Of course h CI)o {.}) = o ( { ~ } ) and h is injective.

    From the sur jec t iv i ty of S i t resul ts that h i s surjective,because for every kEN- i t exists xEN* such that Sex -k , so

    Then we have N, Q) ac N*, Q ) and from the bi ject iv i ty t h i tresul ts cardN-cardN, tha t i s the assert ion a).

    Remarks i) n other proof of Proposition 1 may be made asfollows;

    Let s be the equivalence associated with the function S

    Because S i s a morphism between ~ , Q and :Nt, ell) i t resultsthat Ps is a congruence and so we can define on ~ the operationsP.w and w by

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    CAl Nt / Ps) 2 - - ~ N * Ips C l ( ~ , 9) =:xVtP iC lo : N-/ps) 2 - - ~ N * / p s C l o ( { ~ } ) =1.Moreover, N*/Ps=N and so it i s cons t ruc ted the un ive rsa l

    a lgebra (N, C ) , with Q ={6), . That i s we have a proof fo r (b) , the morphismbeing CI: N - - ~ N * , u (x) =S(x) .

    ( i i ) Propos i t ion 1 i s an argument to cons ider the func t ions~ N * - - ~ N * , s.;fn (k) =minS- (k)~ : N * - - ~ N * , ~ k ) =maxs-(k) ( sec [ 4 ]whose prope r t i e s we sha l l presen t in a fu tu re no te .

    ( i i i ) The graphG = { x ,y) N* XN* I y = Sex)}

    i s a subalgebra o f the un ive rsa l a lgebra (Nt XNt, Q ) , whereQ={{,), 6 o}, with fA : Nt X Nt) 2 _ _ N - XN*, def ined byC l . ~ ; Y l ) , (X2,Y2 ) ) = .x:.. Vfb;. Yl VYa) and C lo: (N*xN*) o -- ..Nx N-, def inedby C A l o ( { ~ } ) = ( 4 t o { { - } ) o ( { ~ } = ( l , l ) .

    Indeed G i s a subalgebra of the univer sa l a lgebra (N*xN*,Q)if fo r every (1C:t, Yl ) , (X2, Y2) G it r e s u l t s fA) .x:.., Yl , (X z, Ya ) G

    a n d f A ) o ( { ~ } ) G . ButC l Xl Yl , (Xa,Y2 = (Xl VdX2 Y1 VY2 ) = (Xl VdX z, S 1C:t) VS (X:l = .x:.. VdXa.

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    and C }()(flJ}) G i f and only i f 1,1) G.Thatis 1 , S (1) GIn fac t the a lgebra ic proper ty i s more complete in the sense

    tha t f :A - - -B i s a morphism between the universa l algebras A, Q )and B, Q) of the some kind i and only i f the graph P of thefunct ional r e la t ion f s a subalgebra of the universa l algebraAxB, Q) .

    Then the importance of remark i i i ) cons i s t in the fac t tha tit s poss ib le to under l ine some proper t ies of the Smarandachefunct ion s t a r t i ng from the above mentioned subalgebra of theun versa l algebra N x N, 0 .

    Reference.1. I . Balacenoiu: Smarandache Numeric Funct ions Smarandache

    Funct ion J , V. 4-5, No.1, 1994), 6-13) .2. I . Purdea, Gh. Pic: Tra ta t de algebra modema, V.1,

    Ed. Acad. R.S.R, Bucure , t i , 1977).3. E. Radescu, N. Radescu, C. Dumitrescu: On the Sumatory

    Funct ion Associa ted to the Smarandache Funct ion SmarandacheFunct ion J . V.4-5, No .1 , 1994), 17-21) .

    4 . F Smarandache: A Funct on in the Number Theory An. Univ .Timi ,oara , Ser . St . Mat., Vol. XVIII, fa sc .1 1980), 79-88).

    Current Addre.s: Universi ty of Craiova, Dept of h . Craiovall )O), Remainia.