Elementos claves en los sistemas de evaluación del desempeño
PLAN DE INCENTIVOS
12. COSTOS DE LA CREACIÓN DEL DEPARTAMENTO DEL TALENTO HUMANO
Purthervjnore, we have Id (g) e C so th a t
s *
^ . ^ s ) 1= defin ed hys' = # y
2(y-|('” (y^y
2^^‘ ” ^ + «
( s -1 ) -tim es where f k , th en g* a lso s a ti s f i e s ^ ^ ) , . . . , (a +i) ' * ^ (n ^ ) , .. ., ( n ^^),(2)>/ \ / \ \ and an in c lu sio n o f the form (*) would not q+1<
Hence i f any of the in te g e rs ^q + 2 '* ""'^k rep laced hy a sm aller in te g e r, th en th e re would he a non-zero elem ent in S(
he tr u e .
S im ila rly , in S^^ th e element
f = Y ^ (Y ^ (...(y ^ y ^ ) ) ...) + . . . (s--2)-tim es s a ti s f i e s C n ^ ,..o ,n -le n g th ( Id (f) ) = (s-2 ) m + p^ ' Hanoe f € .(2 ) which im p lies th a t ^ ( s - l ) ,( 2 )
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Then
hut
la C y .y ^ ) ‘
Id (y^yg) * ),( e + l)
Proceeding s im ila rly one concludes th a t 8/ \ , %
(s-l),(n +2),
has a non-zero element in L. Hence s cannot he rep laced hy a sm aller in te g e r and an in c lu sio n of the type (^) s t i l l he tr u e .
Therefore S i s p o ly n ilp o te n t r e la tiv e to th e sequence
We now g en eralize th is to ;
Theorem 3*4 : Let L = F / ? / \ / \ and S he a
--- ^— C n J , . . . , ( r L J
r 1
tw o-generator suhalgebra of L on g en erato rs Y = ( y^ ’ - "1
Suppose th a t ï|
where p < q < k,. m < n^^^, e.< n^^^. Tlien 8 is poly—- ^
n ilp o te n t r e la tiv e to th e sequence I
Proof : Let
, Id (y^) =
I
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C n ^ , . . . , n ^ ^ l G n g t h ( h ^) = 0, i . e . , h^ < I*(„^) , . . . , („ ^ ), w h io h i m p l i e s th a ,t h ^ h g * %(% ) , . , . , ( n ) , ( e + l ) ' 1 Q. ® ( 2 Y - ) , . . . , (n^),-(e.) , Then hut Furtherm ore , Thus 3 (2 ) ,( '[ n ^ y e ] + 1 ) . { n ^ J , . . . , ( n ^ ) = ^ in L, hut i f any o f th e in te g e rs 2, [n^_^^/e], n^^2, . . . , n ^ i s replaced, hy a sm aller in te g e r th is i s not tru e .T herefore S i s p o ly n ilp o ten t r e la tiv e to th e sequence Î [^q + y ^ ] + ' ^q+2^” *’\ î ’
§ 3# Suhalgehras of a Free P o ly n ilp o ten t Lie Algehra in G eneral
We now g e n e ra lise th e r e s u lts of Chapter 2, § 2 to cover th e p o ly n ilp o te n t c a se . Let L = F / F, \ f \ t where F i s a fre e Lie alg eh ra on fre e gen erato rs X. I f f i s an a r b itr a r y
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elem ent of th en Id (f) ^ B, the a d d itiv e b a sis o f L defined in C hapter 1. The se t B is given a t o t a l order tak in g in to
c o n sid e ra tio n th e follow ing in decreasing order of p r io r ity :
C -le n g th s
C -le n g th s
> • • •
C^ -le n g th s
X -lengths
We w ill use the term "g en eralized length" when comparing elem ents o f B w ith re sp ec t to t h i s ordering#
Lemma 3.2 : Let S be a subalgebra of L and Y be a g en eratin g se t fo i St Let y « Y be such th a t Id (y) i s minimal among th e lead in g term s of th e elem ents o f Y. Suppose y c f o r some q < k , m < n^^^ . Then the in te g e rs m and q are independent of th e choice of the g en eratin g s e t Y and hence th ey are in v a ria n ts o f th e subalgebra S o f L.
P ro o f : Suppose th a t S has a g en erating s e t Y* v/ith an elem ent y^ c having a minimal lead in g term among th e lead in g terms of the elem ents of Y* and assume th a t q = q* and m = m* a re not both tr u e . I f q - q* and m < m*, th e n th e re is an elem ent in S (namely y € Y) which cannot be
expressed in term s of th e generating s e t Y*, since y S f L / % / \ / * Thus we must have m = m*.
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I f q ^ q*, th en a sim ila r c o n tra d ic tio n a r is e s . Therefore th e in te g e rs m and q are- in v a ria n ts o f the suhalgebra S o f L.
Lemma 3.5 : Let S be a sub algebra of L and m , q be as defined b e fo re .. Then one can choose a p a ir of elem ents { ] in Y such th a t
( i) { y . i s lin e a r ly independent modulo L/ / \ / \ | where t > q is th e sm allest such in te g e r, e < n^^^ and i f
t = q, th en e > m is th e sm allest such in te g e r. I f { y . , y . 1- t X J
i s any p a ir of elem ents of Y d is tin c t from [y ^ ,y g ] , and [Yj^jYj'i 4 i s lin e a r ly independent modulo L/ \ ( n ^ ; ,...,( n ^ ,) ,( e * + i ;/ \ / , th en
e ith e r t* > t o r t = t* and e* ^ e .
( i i ) The "g en eralised length" of th e lead in g term o f
y^y^ i s le s s th an o r equal to th a t of any o th e r such p a ir from 3 Y s a tis fy in g ( i ) .
Proof : Let Y be a gen eratin g se t f o r S 'w ith in v a ria n ts q and m as defined b e fo re . This means th a t th e re i s an element y^ in Y whose lead in g term i s minimal among th e lead in g term s o f elem ents of Y and
la. (y,) «-
We prove the lemma by choosing a p a ir ^y^jyg*} which s a ti s f i e s ( i) and ( i i ) .
Suppose th a t y^ i s th e only element in Y whose lead in g term belongs to • Then choose y^ such th a t Id (y^) is minimal among th e lead in g term s o f the elem ents in Y - (y^^ ,
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Then obm ously { * ^ 2 ^ s a ti s f i e s ( i) and ( i i ) .
Now suppose th a t th e re are more th an one elem ent, say ' Y g , . 1 = 1 GY such th a t