topologia

E_{m}[W]...(n+(-m))...[W]E_{n}=E_{n}
E_{m}[M]...(n+(-m))...[M]E_{n}=E_{m}
f( E_{i},Ps[E_{i}] )=E_{i+1}




E_{j}=p^{j}


mcm{p^{m},...(n+(-m))...,p^{n}}=p^{max{m,...(n+(-m))...,n}}
mcd{p^{m},...(n+(-m))...,p^{n}}=p^{min{m,...(n+(-m))...,n}}
mcm{p^{m},...(n+(-m))...,p^{n}}=p^{n}
mcd{p^{m},...(n+(-m))...,p^{n}}=p^{m}


Ps[E_{j}] = (p^{j+1}/p^{j})=p




E_{j}=kp^{j}


mcm{kp^{m},...(n+(-m))...,kp^{n}}=kp^{max{m,...(n+(-m))...,n}}
mcd{kp^{m},...(n+(-m))...,kp^{n}}=kp^{min{m,...(n+(-m))...,n}}
mcm{kp^{m},...(n+(-m))...,kp^{n}}=kp^{n}
mcd{kp^{m},...(n+(-m))...,kp^{n}}=kp^{m}


Ps[E_{j}] = (kp^{j+1}/kp^{j})=p




{15,30,60,120}={15·2^{0},15·2^{1},15·2^{2},15·2^{3}}




E_{j}=j


(1+...(m)...+1)[W]...(n+(-m))...[W](1+...(n)...+1)=(1+...(n)...+1)
(1+...(m)...+1)[M]...(n+(-m))...[M](1+...(n)...+1)=(1+...(m)...+1)
m[W]...(n+(-m))...[W]n=n
m[M]...(n+(-m))...[M]n=m


Ps[E_{j}] = (j+1)+(-1)j=1=(j+1)^{0}


E_{j}=qj


(q+...(m)...+q)[W]...(n+(-m))...[W](q+...(n)...+q)=(q+...(n)...+q)
(q+...(m)...+q)[M]...(n+(-m))...[M](q+...(n)...+q)=(q+...(m)...+q)
qm[W]...(n+(-m))...[W]qn=qn
qm[M]...(n+(-m))...[M]qn=qm


Ps[E_{j}] = q(j+1)+(-1)qj=q=q(j+1)^{0}






E_{j} = ( j(j+1)/2 )


(1+...(m)...+m)[W]...(n+(-m))...[W](1+...(n)...+n) = (1+...(n)...+n)
(1+...(m)...+m)[M]...(n+(-m))...[M](1+...(n)...+n) = (1+...(m)...+m)
( m(m+1)/2 )[W]...(n+(-m))...[W]( n(n+1)/2 ) = ( n(n+1)/2 )
( m(m+1)/2 )[M]...(n+(-m))...[M]( n(n+1)/2 ) = ( m(m+1)/2 )


Ps[E_{j}] = ( (j+1)(j+2)/2 )+(-1)(j(j+1)/2)=j+1


E_{j} = p!( B_{p}(j+1)+(-1)B_{p}(1) )


(1^{p}+...(m)...+m^{p})[W]...(n+(-m))...[W](1^{p}+...(n)...+n^{p}) = (1^{p}+...(n)...+n^{p})
(1^{p}+...(m)...+m^{p})[M]...(n+(-m))...[M](1^{p}+...(n)...+n^{p}) = (1^{p}+...(m)...+m^{p})
( p!( B_{p}(m+1)+(-1)B_{p}(1) ) )[W]...(n+(-m))...[W]( p!( B_{p}(n+1)+(-1)B_{p}(1) ) ) =...
... ( p!( B_{p}(n+1)+(-1)B_{p}(1) ) )
( p!( B_{p}(m+1)+(-1)B_{p}(1) ) )[M]...(n+(-m))...[M]( p!( B_{p}(n+1)+(-1)B_{p}(1) ) ) = ...
...( p!( B_{p}(m+1)+(-1)B_{p}(1) ) )


Ps[E_{j}] = p!( B_{p}((j+1)+1)+(-1)B_{p}(1) )+(-1)p!( B_{p}(j+1)+(-1)B_{p}(1) )=(j+1)^{p}




E_{j}=j!


(1·...(m)...·m)[W]...(n+(-m))...[W](1·...(n)...·n)=(1·...(n)...·n)
(1·...(m)...·m)[M]...(n+(-m))...[M](1·...(n)...·n)=(1·...(m)...·m)
m![W]...(n+(-m))...[W]n!=n!
m![M]...(n+(-m))...[M]n!=m!


Ps[E_{j}] = ( (j+1)!/j! )=j+1


E_{j}=a^{j}!


(1·a·...(m)...·a^{m})[W]...(n+(-m))...[W](1·a·...(n)...·a^{n})=(1·a·...(n)...·a^{n})
(1·a·...(m)...·a^{m})[M]...(n+(-m))...[M](1·a·...(n)...·a^{n})=(1·a·...(m)...·a^{m})
a^{m}![W]...(n+(-m))...[W]a^{n}!=a^{n}!
a^{m}![M]...(n+(-m))...[M]a^{n}!=a^{m}!


Ps[E_{j}] = (a^{j+1}!/a^{j}!)=a^{j+1}


E_{j} = {a_{1},...,a_{j}}


{a_{1},...,a_{m}}[W]...(n+(-m))...[W]{a_{1},...,a_{n}}={a_{1},...,a_{n}}
{a_{1},...,a_{m}}[M]...(n+(-m))...[M]{a_{1},...,a_{n}}={a_{1},...,a_{m}}
E_{m}[W]...(n+(-m))...[W]E_{n}=E_{n}
E_{m}[M]...(n+(-m))...[M]E_{n}=E_{m}




Ps[E_{j}] = {a_{1},...,a_{j+1}} --- {a_{1},...,a_{j}}={a_{j+1}}


Ad[ {x€R : d_{j}(x,c) < s_{j}} ] = {x€R : d_{j}(x,c) [< s_{j}} & s_{j+1} > s_{j} >] 0
Int[ {x€R : d_{j}(x,c) [< s_{j}} ] = {x€R : d_{j}(x,c) < s_{j}} & s_{j+1} > s_{j} >] 0
Fr[ {x€R : d_{j}(x,c) [< s_{j}} , {x€R : d_{j}(x,c) < s_{j}} ]={s_{j}}


{x€R : d_{m}(x,c) [< s_{m}}[W]...(n+(-m))...[W]{x€R : d_{n}(x,c) [< s_{n}} = {x€R : d_{n}(x,c) [< s_{n}}
{x€R : d_{m}(x,c) [< s_{m}}[M]...(n+(-m))...[M]{x€R : d_{n}(x,c) [< s_{n}} = {x€R : d_{n}(x,c) [< s_{m}}
{x€R : d_{m}(x,c) [< s_{m}} [<< ...(n+(-m))... [<< {x€R : d_{n}(x,c) [< s_{n}}


{x€R : d_{m}(x,c) < s_{m}}[W]...(n+(-m))...[W]{x€R : d_{n}(x,c) < s_{n}} = {x€R : d_{n}(x,c) < s_{n}}
{x€R : d_{m}(x,c) < s_{m}}[M]...(n+(-m))...[M]{x€R : d_{n}(x,c) < s_{n}} = {x€R : d_{m}(x,c) < s_{m}}
{x€R : d_{m}(x,c) < s_{m}} [<< ...(n+(-m))... [<< {x€R : d(x,c) < s_{n}}


Ps[ {x€R : d_{j}(x,c) [< s_{j}} ]={ x€R : d_{j+1}(x,c)+(-1)d_{j}(x,c) [< s_{j+1}+(-1)s_{j} }
Ps[ {x€R : d_{j}(x,c) < s_{j}} ]={ x€R : d_{j+1}(x,c)+(-1)d_{j}(x,c) < s_{j+1}+(-1)s_{j} }


Ad[ ((-1)a_{j},a_{j})_{R} ]=[(-1)a_{j},a_{j}]_{R} & a_{j} < a_{j+1}
Int[ [(-1)a_{j},a_{j}]_{R} ]=((-1)a_{j},a_{j})_{R} & a_{j} < a_{j+1}
Fr[ [(-1)a_{j},a_{j}]_{R} , ((-1)a_{j},a_{j})_{R} ]={ (-1)a_{j},a_{j} }


Ps[ [(-1)a_{j},a_{j}]_{R} ]={ x€R : (-1)a_{j+1} [< x [< (-1)a_{j} or a_{j} [< x [< a_{j+1} }
Ps[ ((-1)a_{j},a_{j})_{R} ]={ x€R : (-1)a_{j+1} < x < (-1)a_{j} or a_{j} < x < a_{j+1} }