Temporal Becoming and the algebra of time

7/30/2019 Temporal Becoming and the algebra of time

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1


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e

e

e


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G

e e

e e G


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e e G

e


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3


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M , , M

x X T x

x M T x x(X, X )

x yx

x xx M

x

x

X T x x

x MY T x X (X, Y ) = 0

x x x

xx

xx


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xM

xx


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(X 1, O1, T 1) W 1 O1

(X 2, O2, T 2) W 2 O2 W 1 W 2T 1 = 0 (X 2, O2, T 2)


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O1 X 2 T 1 P W 1 W 2 O2 = BW 2 B B P W 2

P T 1 = 0 W 1 W 2 W 1P W 2 W 1 W 2

W 1 P T 1 = 0 W 1P 1 W 1 W 2

W 1 W 2 W 1 P 1

P W 2W 2 W 1 P

T 1 = 0W 1

P

W 1

P W 1

P W 2W 2 T p W 1

P P P W 1W 1

W 2W 1 P O1


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W 2

T 1 = 0 O2 W 2O2 W 1

W 2 P O2W 2 W 1 W 1 P O2

W 2 W 2 W 1P O1

W 2W 1 O1 P W 2

W 2 BW 1 t

(X 1, O2, T 1)W 1 W 2

W 2

P tW 1 W 1

W 1 P

W 2O1

P W 2O2 P

W 1 W 2W 1 P P O2W 1

W 1W 2 W 1


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R

R

R

R

R


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R

RR


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H

W i W W W j W W i W j


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W i W j W iW j

W i

W (R1, R 2) R1 R2W j W W i

W j R1R2 R1 R2 W i

W j W i W j R1R2

RM Rab a

b b a

R

Rab (a, b) ab


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(x, y)Rxy xy

a b ab a b

a b c a b b

c a c

M

1 ,...,n L T

i i T T i


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RRba

baR


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R4

M R3 T

M

x


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x

x

xx

x

x

x

x

x

A S x Ax A A

x A Ax A A

A


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x

S E S

E

xx

x

M , ,S ( , )

U V

S ( , ) O1O2

M , ,

S ( , )


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S ( , )

S ( , )

x


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P

P P

P P

P P

x y z xy z

x x


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x

S ( , )


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M


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4


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x i x j x i x j

x i x jx i x i x j x i

x j

x i x j x i x j

x i x j x i = x j x i x j

x i x j x i = x j x i x j

x i x j x i = x j

x i x i x j

x ix i x j x i x j x j


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x i x i x jx i

x i x j x i x j x j

x i x j x i x j x j = x i

(x i )ni=1 i = 1 ,...,n 1 x ix i+1 xn x i

(x i )+ i=1 i = 1 , ..., + x i x i+1


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c d

c/d


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n/n = 1 n = 0n : mn = m

n : m n = m


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T U 1 T U 2

T U 1


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s0, s1,...,s n T U 1 S 0, S 1,...,S nT U 2 T U 2

T U 1s i s j

S i S j

T U 1T U 2 T U 1

T U 1T U 1

T U 2 T U 2

T U 1


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1

2


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5

n

R

Z+ Z R R+

DS

DS = ( M, (gt ) tT ),

M

T Z, Z+ , R, R+

(gt )tT M T


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x M t, v T

g0(x) = x

gt + v (x) = gt (gv (x)) .

M T

gt

t T t x M gt gt (x) tx

R

a1 , a 2 ,...,a n ,... r n 1 n

a n = a1 r n 1

a i {a i }iZ + { 0} n Z+

gn (a i ) = a i r n .

M = {a i }iZ + { 0}

Z+

(gn )n Z + M M Z+

a i M n N

gn (a i ) = a i r n = a1 r i 1 r n = a1 r i 1+ n = a i + n M ;

a i M n, m Z+

g0 (a i ) = a i r 0 = a i

gn + m (a i ) = a i r n + m = a i r n r m = gn (gm (a i )) .

a1 , a 2 ,...,a n ,...

t T t0 T t0 x M

t + t0 gt (x)

0v + t t


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v

L DS L

DS L = ( M, (gt )tT ),

M

L = ( T, +) 0

(gt )tT M T

x M t, v T

g0(x) = x,

gt + v (x) = gt (gv (x)) .

Z+ Z RR+ L


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L = ( T, +) DS L

DS L = ( M, (gt )tT ),

M

(gt )tT M M T

DS L 1 = ( M 1, (gt 1 )t 1T 1 ) L1 = ( T 1, +)

DS L 2 = ( M 2, (gt 2 )t 2T 2 ) L2 = ( T 2,) f DS L 1 DS L 2

: T 1 T 2 L1 L2


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f : M 1 M 2 M 1 M 2 x1 M 1 t1 T 1

f (gt 1 (x1)) = g(t 1 )(f (x1)) .

DS L 1 L1 DS L 2L2 DS L 2 DS L 1 f

f DS L 2 DS L 1

DS L 1 = ( M 1 , (gt 1 ) t 1 T 1 ) L1 = ( T 1 , +)

DS L 2 = ( M 2 , (gt 2 ) t 2 T 2 ) L2 = ( T 2 ,)

DS L 3 = ( M 3 , (gt 3 ) t 3 T 3 ) L3 = ( T 3 , )

DS L 1 1 , 1 : T 1 T 1 T 1 f 1 , 1 : M 1 M 1M 1 f 1 , 1 1 , 1 DS L 1 DS L 1

1 , 1 L1

f 1 , 1

x1 M 1 t1 T 1

f 1 , 1 (gt 1 (x1 )) = f 1 , 1 (g 1 , 1 ( t 1 ) (x1 )) = g 1 , 1 ( t 1 ) (x1 ) = g 1 , 1 ( t 1 ) (f 1 , 1 (x1 )) .


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DS L 1 DS L 1

DS L 1 DS L 2 1 , 2 : T 1 T 2 L1 L2 f 1 , 2 : M 1 M 2 f 1 , 2 1 , 2 DS L 1 DS L 2 1 , 2 f 1 , 2(1 , 2 ) 1 : T 2 T 1 (f 1 , 2 ) 1 : M 2 M 1 (f 1 , 2 ) 1 (1 , 2 ) 1 DS L 2 DS L 1

(1 , 2 ) 1 L2 L1

(f 1 , 2 ) 1 f 1 , 2

x2 M 2 t2 T 2

g( 1 , 2 ) 1 ( t 2 ) (( f 1 , 2 ) 1 (x2 )) = ( f 1 , 2 ) 1 (f 1 , 2 (g( 1 , 2 )

1 ( t 2 ) (( f 1 , 2 ) 1 (x2 ))))

= ( f 1 , 2 ) 1 (g 1 , 2 (( 1 , 2 ) 1 ( t 2 )) (f 1 , 2 (( f 1 , 2 ) 1 (x2 ))))

= ( f 1 , 2 ) 1 (gt 2 (x2 ))

DS L 2 DS L 1

DS L 1 DS L 2 DS L 2 DS L 31 , 2 : T 1 T 2 L1 L2 2 , 3 : T 2 T 3 L2 L3 f 1 , 2 : M 1 M 2f 2 , 3 : M 2 M 3 f 1 , 2 1 , 2 DS L 1 DS L 2 f 2 , 3 2 , 3 DS L 2

DS L 3 (2 , 3 1 , 2 ) : T 1 T 3 f 2 , 3 f 1 , 2 : M 1 M 3 f 2 , 3 f 1 , 2(2 , 3 1 , 2 ) DS L 1 DS L 3

2 , 3 1 , 2 L1 L3

f 2 , 3 f 1 , 2

x1 M 1 t1 T 1

f 2 , 3 (f 1 , 2 (gt 1 (x1 ))) = f 2 , 3 (g 1 , 2 ( t 1 ) (f 1 , 2 (x1 ))) = g 2 , 3 ( 1 , 2 ( t 1 )) (f 2 , 3 (f 1 , 2 (x1 ))) .

DS L 1 DS L 3

R A n A n 1 a1, b1, ...a n , bn A (a1,...a n )

R(a1, b1), ...R (an , bn ) (b1,...bn )

DS L 1 L1 DS L 2L2 DS L 1 L1 DS L 1

DS L 2 DS L 2 L2

DS L 1 = ( M 1 , (gt 1 ) t 1 T 1 ) DS L 2 = ( M 2 , (gt 2 ) t 2 T 2 )L1 = ( T 1 , +) L2 = ( T 2 ,) DS L 1 DS L 2

: T 1 T 2 f : M 1 M 2 f DS L 1 DS L 2 DS L 1L1 L1 L2

0 T 1 L1 (0) L2

M 2

(gt ) tT M T


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f (x1 ) M 2 (t1 ), (v1 ) T 2

g (0) (f (x1 )) = f (g0 (x1 )) = f (x1 )

g ( t ) ( v ) = g ( t + v ) (f (x1 )) = f (gt + v (x1 )) = f (gt (gv (x1 ))) = g ( t ) (f (gv (x1 ))) = g ( t ) (g ( v ) (f (x1 ));

f x2 M 2 x1 M 1f t 2 T 2 t 1 T 1

x2 M 2 t2 , v2 T 2

DS L 2

DS L 1 DS L 2L1 L2

DS L 1

DS L 1 L1

DS L 1 DS L 1 DS L 2 DS L 2

DS L 1 DS L 2 L1 L2DS L 1

DS L 2


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G

G = ( X,A,, ),

G A

A X

X A a A (a)

a (a)

tDS L

DS L = ( M, (gt )tT ) L = ( T, +)DS L G(DS L )

G(DS L ) = ( X,A,, )

X = M

A = a : t T x M, a = ( x,t ,g t (x))

: A X A M a A (a)a

: A X A M a A (a)a


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G( DS L ) = ( X,A,, ) DS L = ( M, (gt ) tT )L = ( T, +) M x M y M t T y = gt (x) X A

A X G(DS L )

L = ( T, +) 0 DS L

DS L = ( M, (gt )tT ),

M

(gt )tT M M T

x M t, v T

g0(x) = x

gv+ t (x) = gv (gt (x))


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G( DS L ) = ( X,A,, ) DS L Lx (idx )xX

DS L

DS L = ( M, (gt ) tT ) L = ( T, +) 0DS L G( DS L )

G( DS L ) = ( G, (idx )xX , )

G = ( X,A,, ) = G( DS L )

(idx )xM x M

(idx ) = ( x, 0, g0(x));

a, b A a = ( x,t ,g t (x))b = ( gt (x),v,g v (gt (x))) t, v T

b a = ( x, v + t, g v+ t (x)));

b a

DS L = ( M, (gt )tT ) L = ( T, +) AG( DS L ) = ( X,A,, ) a = ( x,t ,g t (x))

b = ( gt (x),v,g v (gt (x))) A (a) = (b)b a = ( x, v + t, g v+ t (x))) b a A

Lgv+ t DS L

x gv+ t X v + t T b aA

G

G = ( G, (idx )xX , )


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G = ( X,A,, )

(idx )xX X x X

(idx ) = x (idx ) = x;

Aa, b A (a) = (b) b a

(b a) = a (b a) = (b).

DS L = ( M, (gt ) tT ) L = ( T, +) 0 G( DS L ) =(G, (id x )xX , ) DS L G = ( X,A,, )

G

(id x )xM x X = M

(id x ) = (x, 0, g0 (x)) = x,

(id x ) = (x, 0, g0 (x)) = g0 (x) = x;

a = ( x,t ,g t (x))b = ( gt (x), v, g v (gt (x))) A

(b a) = (( x, v + t, g v + t (x))) = x = (a),

(b a) = (( x, v + t, g v + t (x))) = gv + t (x) = gv (gt (x)) = (b),

G( DS L )

G = ( G, (id)xX , ) G = ( X,A,, )a,b,c A (a) = (b) (b) = (c)

c (b a) = ( c b) a.

G( DS L )DS L L

DS L = ( M, (gt

) tT ) L = ( T, +) 0 G( DS L ) =(G, (id)xX , ) DS L G = ( X,A,, )


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a = ( x,t ,g t (x)) a = x tgt (x) a,b,c A a = x t gt (x) b = gt (x) v gv (gt (x))c = gv (gt (x)) u gu (gv (gt (x))) G( DS L )

L +

(c b) a = [ gv (gt (x)) u gu (gv (gt (x))) gt (x) v gv (gt (x))] x t gt (x)

= gt (x) u + v gu (gv (gt (x))) x t gt (x)

= x( u + v )+ t

gu (gv (gt (x)))

= xu +( v + t )

gu (gv (gt (x)))

= gv (gt (x)) u gu (gv (gt (x))) x v + t gv (gt (x))

= gv (gt (x)) u gu (gv (gt (x))) [gt (x) v gv (gt (x)) x t gt (x)]

= c (b a);

G( DS L )

L u,v, t T

u + ( v + t) = ( u + v) + t

x M

xu +( v + t )

gu +( v + t ) (x) = x( u + v )+ t

g( u + v )+ t (x);

c (b a) = ( c b) a,

G( DS L ) G( DS L ) L


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n

tt x

DS L = ( M, (gt )tT ) L = ( T, +) x M DS L x gx

gx : T M t T

gx = eval (gt , x),

t T x M

eval (gt

, x) = gt

(x).

DS L = ( M, (gt )tT ) L = ( T, +)x M orb(x)

orb(x)def = y M : t T (y = gt (x)) .

x gx orb(x)

x

DS L 1 = ( M 1, (gt 1 )t 1T 1 ) L1 = ( T 1, +)DS L 2 = ( M 2, (gt 2 )t 2T 2 ) L2 = ( T 2,) f : M 1

M 2 DS L 1 DS L 2 x1, y1 M 1 y1 orb(x1)f (y1) orb(f (x1))

DS L 1 = ( M 1 , (gt 1 ) t 1 T 1 ) DS L 2 = ( M 2 , (gt 2 ) t 2 T 2 ) L1 = ( T 1 , +) L2 =


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(T 2 ,) : T 1 T 2 L1 L2 f : M 1 M 2 DS L 1 DS L 2 x1 , y1 M 1 y1 orb(x1 ) t1 T 1

gt 1 (x1 ) = y1

f (gt 1 (x1 )) = f (y1 )

g ( t 1 ) (f (x1 )) = f (y1 ),

f (y1 ) orb(f (x1 ))f 1 : M 2 M 1 1 DS L 2 DS L 1

DS L LDS x L L x

DS L x

DS L = ( M, (gt )tT ) L = ( T, +)x M DS x L = ( orb(x), (gt |orb (x ) )tT ) L

DS = ( M, (gt ) tT ) L = ( T, +) 0 x M

orb(x) x

T DS

(gt |orb ( x ) ) tT orb(x) T y orb(x)t T v T

gt |orb ( x ) (y) = gt |orb ( x ) (gv (x)) = gt |orb ( x ) (gv |orb ( x ) (x)) = gt + v |orb ( x ) (x) orb(x);

y orb(x) t T

g0 |orb ( x ) (y) = g0 (y) = y

gt + v |orb ( x ) (y) = gt + v (y) = gt (gv (y)) = gt |orb ( x ) (gv |orb ( x ) (y))

DS x L = ( orb(x), (gt |orb ( x ) ) tT ) L


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DS L = ( M, (gt )tT ) L = ( T, +)x, y M y orb(x) orb(y) orb(x)

DS L = ( M, (gt ) tT ) L = ( T, +) 0 x, y M

y orb(x) v T

gv (x) = y;

t T gt (y) orb(y)

gt (y) = gt (gv (x)) orb(x),

orb(y) orb(x)

orb(y) orb(x) t T gt (y) orb(y) v T gv (x) orb(x)

gt (y) = gv (x);

t = 0gv (x) = g0 (y) = y,

y orb(x)

x yx y

DS L = ( M, (gt )tT ) L = ( T, +)x,y,z M z orb(x) orb(y) orb(z) orb(x) orb(z) orb(y)

DS L = ( M, (gt ) tT ) L = ( T, +) x,y,z M z orb(x)orb(y)z orb(x) z orb(y) orb(z ) orb(x) orb(z ) orb(y)

DS L = ( M, (gt )tT ) L = ( T, +) rx, y M r = orb(x)

orb(x) orb(y),

orb(y) orb(x),

orb(x) orb(y) = .


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DS L = ( M, (gt )tT ) L = ( T, +) r sr s r s


M 2 DS L 1 DS L 2 x1 orb(x) orb(f (y1))

DS L 1 = ( M 1 , (gt 1 ) t 1 T 1 ) DS L 2 = ( M 2 , (gt 2 ) t 2 T 2 ) L1 = ( T 1 , +) L2 =(T 2 ,) : T 1 T 2 L1 L2 f : M 1 M 2

DS L 1 DS L 2 x1 M 1 orb(x1 ) y1orb(x1 ) orb(y1 ) orb(y1 ) orb(x1 ) orb(x1 ) orb(y1 ) =

orb(f (x1 )) orb(f (y1 )) ,

orb(f (y1 )) orb(f (x1 ))

z 1 orb(x1 ) orb(y1 )f (z 1 ) orb(f (x1 )) orb(f (y1 ));

orb(f (x1 )) f 1 : M 2 M 1 1 DS L 2 DS L 1

DS L = ( M, (gt )tT ) L = ( T, +) 0x M x t T

t = 0 , gt (x) = x.

DS L = ( M, (gt )tT ) L = ( T, +) 0 r

x M r = orb(x) x


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x

DS L = ( M, (gt)tT ) L = ( T, +) 0

t T t orb(x)

t = 0 ,

gt (x) = x.

x

DS L 1 = ( M 1, (gt 1 )t 1T 1 ) L1 = ( T 1, +)DS L 2 = ( M 2, (gt 2 )t 2T 2 ) L2 = ( T 2,) : T 1 T 2

L1 L2 f : M 1 M 2 DS L 1 DS L 2x1 M 1 t1 T 1 orb(x1) (t1)

orb(f (x1))

DS L 1 = ( M 1 , (gt 1 ) t 1 T 1 ) L1 = ( T 1 , +) 0 DS L 2 =(M 2 , (gt 2 ) t 2 T 2 ) L2 = ( T 2 ,) : T 1 T 2L1 L2 f : M 1 M 2 DS L 1 DS L 2 x1 M 1 t1 T 1 t1

orb(x1 ) t 1 = 0

f (x1 ) = f (gt 1 (x1 )) = g t 1 (f (x1 )) ,

(t1 ) T 2 (t1 ) orb(f (x1 ))

f 1 : M 2 M 1 1 DS L 2 DS L 1

DS L = ( M, (gt )tT ) L = ( T, +) rr r y r orb(y)

DS L 1 = ( M 1, (gt 1 )t 1T 1 ) L1 =(T 1, +) DS L 2 = ( M 2, (gt 2 )t 2T 2 ) L2 = ( T 2,)

f : M 1 M 2 DS L 1 DS L 2 x1 M 1 orb(x1)orb(f (x1))


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DS L 1 = ( M 1 , (gt 1 ) t 1 T 1 ) L1 = ( T 1 , +) 0 DS L 2 =(M 2 , (gt 2 ) t 2 T 2 ) L2 = ( T 2 ,) : T 1 T 2L1 L2 f : M 1 M 2 DS L 1 DS L 2 orb(x1 )

y1 orb(x1 ) t1 T 1 { 0} v1 T 1 { 0}

gt 1 (x1 ) = x1

gv 1 (y1 ) = y1 ;

f (y1 ) orb(f (x1 )) (t 1 ) T 2 (v1 ) T 2

g ( t 1 ) (f (x1 )) = f (x1 )

g ( v 1 ) (f (y1 )) = f (y1 ),

(v1 ) L2 orb(f (x1 ))f 1 : M 2 M 1 1 DS L 2 DS L 1

x x

DS L = ( M, (gt )tT ) L = ( T, +) rr

DS L 1 = ( M 1, (gt 1 )t 1T 1 ) L1 =(T 1, +) DS L 2 = ( M 2, (gt 2 )t 2T 2 ) L2 = ( T 2,)f : M 1 M 2 DS L 1 DS L 2 x1 M 1 orb(x1)

orb(f (x1))

DS L 1 = ( M 1 , (gt 1 ) t 1 T 1 ) L1 = ( T 1 , +) 0 DS L 2 =(M 2 , (gt 2 ) t 2 T 2 ) L2 = ( T 2 ,) : T 1 T 2L1 L2 f : M 1 M 2 DS L 1 DS L 2 orb(x1 )

orb(f (x1 )) orb(f (x1 ))f 1 : M 2 M 1 1 DS L 2

DS L 1

y y

DS L = ( M, (gt )tT ) L = ( T, +)x, y M y orb(x) orb(y) orb(x)

DS L = ( M, (gt ) tT ) L = ( T, +) 0 x, y M y orb(x) orb(y) t T { 0} gt (x) = x

orb(x) t T { 0} gt (x) = xorb(x)


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DS L = ( M, (gt )tT ) L = ( T, +)x, y M y orb(x) orb(x) orb(y)

DS = ( M, (gt ) tT ) L = ( T ; +) x, y M y orb(x)orb(y) z orb(y) orb(x) orb(z )

orb(x) orb(x)orb(y) orb(y)

x

tDS L = ( M, (gt )tT ) L = ( T, +) 0

x M t T { 0} t x F t (x)

F t (x)def = y M : y = gt (x) .

t xt x x

t

DS L = ( M, (gt )tT ) L = ( T, +) 0x M x F (x)

F (x)def

=tT { 0}

F t (x) = y M : t T { 0} , y = gt (x) .

xx F (x) orb(x) x

x

t


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tDS L = ( M, (gt )tT ) L = ( T, +) 0

x M t T { 0} t x P t (x)

P t (x) def = y M : x = gt (y) .

DS L = ( M, (gt )tT ) L = ( T, +) 0x M x P (x)

P (x)def =

tT { 0}

P t (x) = y M : t T { 0} , x = gt (y) .

xt t x

t

tx t x

DS L = ( M, (gt

)tT ) L = ( T, +)x, y M t T { 0} x P t (y) y F t (x)

DS L = ( M, (gt ) tT ) L = ( T, +) 0 x, y M t T { 0}

x P t (y) t T { x}

gt (x) = y;

t y F t (x)

y F t (x) t T { x}

gt (x) = y;

t x P t (y)

DS L = ( M, (gt )tT ) L = ( T, +)x, y M x P (y) y F (x)

DS L = ( M, (gt ) tT ) L = ( T, +) 0 x, y M


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x P (y) t T { 0} x P t (y)y F t (x) y F (x) y F (y)t T { 0} y F t (x) x P t (y)

x P (y)

DS L = ( M, (gt )tT ) L = ( T, +)x M orb(x) x F (x)

DS L = ( M, (gt ) tT ) L = ( T, +) 0 x M

orb(x) t T { 0}

gt

(x) = x;

t x F t (x) x F (x) x F (x)

x tT { 0}

F t (x),

t T { 0} x F t (x) t

gt (x) = x;

orb(x)

DS L

= ( M, (gt )tT

) L = ( T, +)x M orb(x) x P (x)

DS L = ( M, (gt ) tT ) L = ( T, +) 0 x M orb(x) x F (x)

orb(x) x P (x)

DS L = ( M, (gt ) tT ) L = ( T, +) x M

t T gt (x) = x.


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DS L 1 = ( M 1, (gt 1 )t 1T 1 ) L1 =(T 1, +) DS L 2 = ( M 2, (gt 2 )t 2T 2 ) L2 = ( T 2,)f : M 1 M 2 DS L 1 DS L 2 x1 M 1 x1f (x1)

DS L 1 = ( M 1 , (gt 1 ) t 1 T 1 ) L1 = ( T 1 , +) 0 DS L 2 =(M 2 , (gt 2 ) t 2 T 2 ) L2 = ( T 2 ,) : T 1 T 2L1 L2 f : M 1 M 2 DS L 1 DS L 2 x1 M 1 x1

t1 T 1g ( t 1 ) (f (x1 )) = f (gt 1 (x1 )) = f (x1 ),

f (x1 )f 1 : M 2 M 1 1 DS L 2 DS L 1

DS L = ( M, (gt )tT ) L = ( T, +)x M x

F (x) = {x} .

DS L = ( M, (gt ) tT ) L = ( T, +) 0 x M

x t T gt (x) = x

t T { 0}

F t (x) = y M : gt (x) = y = {x} ,

F (x) =tT { 0}

F t (x) = {x} .

F (x) = {x}

{x} = tT { 0} F t

(x) = tT { 0} y M : gt

(x) = y ,

t T { 0}gt (x) = x;

g0 (x) = x,

t T gt (x) = x;

x

DS L = ( M, (gt

)tT ) L = ( T, +)x M x orb(x) = {x}


84/169

DS = ( M, (gt ) tT ) L = ( T, +) 0 x M

xF (x) = {x} ;

orb(x) =tT

y M : gt (x) = y = F (x) y M : g0 (x) = y = {x} {x} = {x} .

orb(x) = {x}

{x} = y M : t T, g t (x) = y ,

t T gt (x) = x.

DS L = ( M, (gt )tT ) L = ( T, +)0 x M x t T { 0} orb(x)

DS L

= ( M, (gt )tT

) L = ( T, +) 0 x M t T { 0} gt (x) = x t

orb(x)

DS L

= ( M, (gt )tT

) L = ( T, +) 0x M y M t T { 0}

gt (y) = x.

DS L = ( M, (gt )tT ) L = ( T, +)x M x P (x) =

DS L = ( M, (gt ) tT ) L = ( T, +) 0 x M

x

P (x) = y M : t T { 0} , g t (y) = x) = .


85/169

P (x) = t T { 0} y M

gt (y) = x;

x

DS L 1 = ( M 1, (gt 1 )t 1T 1 ) L1 =(T 1, +) DS L 2 = ( M 2, (gt 2 )t 2T 2 ) L2 = ( T 2,)f : M 1 M 2 DS L 1 DS L 2 x1 M 1 x1

f (x1)

DS L 1 = ( M 1 , (gt 1 ) t 1 T 1 ) L1 = ( T 1 , +) 0 DS L 2 =(M 2 , (gt 2 ) t 2 T 2 ) L2 = ( T 2 ,) : T 1 T 2L1 L2 f : M 1 M 2 DS L 1 DS L 2 x1 M 1 x1

t1 T 1 { 0} y1 M 1gt 1 (y1 ) = x1

f f (y1 ) M 2 (t1 ) T 2 { (0) }

g ( t 1 ) f (y1 ) = f (gt 1 (y1 )) = f (x1 );

f (x1 ) f 1 : M 2 M 1

1

DS L 2 DS L 1

y zx z y


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DS L = ( M, (gt )tT )L = ( T, +) x,y,z M y orb(x) z orb(x) orb(z)orb(y) =

DS L = ( M, (gt ) tT ) L = ( T, +) x,y,z M y orb(x) z orb(x) t, v T

gt (x) = y

gv (x) = z ;

gt (z ) = gt (gv (x)) = gt + v (x) = gv + t (x) = gv (gt (x)) = gv (y),

gt (z ) = gv (y) orb(z ) gt (z ) = gv (y) orb(y) orb(z ) orb(y) =

DS L = ( M, (gt )tT )L = ( T, +) 0 x, y M t T { 0} y orb(x) t

x t y

DS L = ( M, (gt ) tT ) L = ( T, +) 0x, y M y orb(x) v T

gv (x) = y,

t T { 0} xgt (x) = x;

gt (y) = gt (gv (x)) = gt + v (x) = gv + t (x) = gv (gt (x)) = gv (x) = y,

t y

DS L = ( M, (gt )tT )L = ( T, +) x, y M orb(x) y orb(x) orb(y)

DS L = ( M, (gt ) tT ) L = ( T, +) 0x, y M orb(x) t T { 0} t x

t y orb(y)


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DS L = ( M, (gt )tT )L = ( T, +) x, y M y orb(x) orb(y) orb(x)

DS L = ( M, (gt ) tT ) L = ( T ; +) x, y M y orb(x) orb(y)

orb(x) orb(x)z orb(x)

w orb(z ) w orb(y) orb(w)orb(y) orb(x)

DS L = ( M, (gt )tT )L = ( T, +) x, y M y orb(x) orb(x) orb(y)

DS L = ( M, (gt ) tT ) L = ( T ; +) x, y M y orb(x) orb(x) orb(y)

orb(y) orb(x)

DS L = ( M, (gt )tT )L = ( T, +) x, y M orb(x) orb(y)

DS L = ( M, (gt ) tT ) L = ( T, +) x, y M orb(x)orb(y) z M z orb(x) orb(y) z orb(x)

z orb(y) orb(x) orb(z )orb(y)


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6

DS L = ( M, (gt )tT ) L = ( T, +)t T gt


89/169

t

DS L = ( M, (gt )tT ) L = ( T, +)t T gt

DS L = ( M, (gt ) tT ) L = ( T, +) 0DS L x M t T { 0} y M

gt (y) = x,

gt DS L

DS L = ( M, (gt )tT ) L = ( T, +)t T gt

y t x

x y


90/169


M 2 DS L 1 DS L 2

DS L 1 DS L 2

DS L 1 DS L 2

DS L 1 DS L 2


DS L 1 DS L 2

x2 , y2 M 2 t 2 T 2 f x1 , y1 M 1t1 T 1 x2 = f (x1 ) y2 = f (y1 ) t2 = (t1 ) DS L 1

gt 2 (x2 ) = gt 2 (y2 )

g ( t 1 ) (f (x1 )) = g ( t 1 ) (f (y1 ))

f (gt 1 (x1 )) = f (gt 1 (y1 ))

gt 1 (x1 ) = gt 1 (y1 )

x1 = y1

f (x1 ) = f (y1 )

x2 = y2 ,

DS L 2

x2 M 2 t2 T 2 f x2 = f (x1 ) x1 M 1 t 2 = (t1 )t1 T 1 DS L 1 y1 M 1 y2 = f (y1 ) M 2

gt 1 (y1 ) = x1

x2 = f (x1 ) = f (gt 1 (y1 )) = g ( t 1 ) (f (y1 )) = gt 2 (y2 );

DS L 2

DS L 1DS L 2

DS L2

f 1 1 DS L 2DS L 1


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DS L = ( M, (gt ) tT ) L = ( T, +)x M t T r T

gr (gt (x)) = x.


M 2 DS L 1 DS L 2 DS L 1 DS L 2

DS L 1 = ( M 1 , (gt 1 ) t 1 T 1 ) DS L 2 = ( M 2 , (gt 2 ) t 2 T 2 ) L1 = ( T 1 , +) L2 =

(T 2 ,) : T 1 T 2 L1 L2 f : M 1 M 2 DS L 1 DS L 2 DS L 1 t1 T 1 x1 M 1 v1 T 1

gv 1 (gt 1 (x1 )) = gv 1 + t 1 (x1 ) = x1 .

x2 M 2 t2 T 2

x2 = f (x1 )

t 2 = (t1 )

v2 = (v1 ) T 2

f (x1) = f (gv 1 + t 1 (x

1))

= g ( v 1 + t 1 ) (f (x1 ))

= g ( v 1 ) ( t 1 ) (f (x1 ))

= g ( v 1 ) (g ( t 1 ) (f (x1 ))) .

f x2 M 2 t 2 T 2 x1 M 1 t1 T 1x2 t2 DS L 2

f 1 : M 2 M 1 1 DS L 2DS L 1


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DS L = ( M, (gt )tT ) L = ( T, +)x M

P (x) = F (x).

DS L = ( M, (gt ) tT ) L = ( T, +) x M

DS L

y F (x) t T { 0} r T

gt (x) = y

gr (y) = gr (gt (x)) = x.

r = 0 y P (x) r = 0 y = x gt (y) = x y P (x)

y P (x) t T { 0} r T

gt (y) = x

y = gr (gt (y)) = gr (x).

r = 0 y F (x) r = 0 y = x gt (y) = x y F (x)

F (x) P (x) P (x) F (y)

F (x) = P (x).

F (x) = P (x) t T { 0} y P (x) r T

gt (x) = y

x = gr (y) = gr (gt (x)) .

t = 0 DS L

DS L = ( M, (gt ) tT ) L = ( T, +) x M DS LP (x) = F (x) F (x)

P (x) xx DS L


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DS L = ( M, (gt )tT ) L = ( T, +)x, y M orb(x) orb(y) = orb(y) = orb(x)

DS L = ( M, (gt ) tT ) L = ( T, +)

DS L y, x M y orb(x) t T r T { 0}

gt (x) = y

gr (y) = gr (gt (x)) = x,

x orb(x) orb(x) orb(y) orb(y) orb(x) orb(x) =orb(y) x, y M orb(x) orb(y) = z M z orb(x)z orb(y) orb(x) = orb(z ) = orb(y)

DS L x, y M t T r T

gt(x) = y

gr (y) = gr (gt (x)) = x,

orb(x) = orb(y) y orb(x) orb(x) orb(y) =

DS L = ( M, (gt ) tT ) L = ( T, +) DS Lx, y M orb(x) orb(y) orb(y) orb(x) orb(x) orb(y) = DS L

L = ( T, +) T = {0, 1, 2, 3} +

M = {x i }4i =1 t T x i M gt (x i )

L + T 0

DS L = ( M, (gt ) tT ) L

M


94/169

+ 0 1 2 30 0 1 2 31 1 1 2 32 2 1 2 3

3 3 1 2 3+ +

g0 g1 g2 g3

x1 x1 x2 x1 x3x2 x2 x2 x1 x3x3 x3 x2 x1 x3

t (gt ) t T

(gt ) tT M T

x i M t, v T

DS L

DS L t T { 0} gt

DS L t T { 0} gt

x Z n Z+ gn (x) n x

DS L = ( Z, (gn )n Z + ) L = ( Z+ , +)


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Z(gn )n Z + Z Z+

x Z n, m Z+

g0

(x) = x + 0 = xgn + m (x) = x + ( n + m ) = ( x + n) + m = gm (gn (x)) .

DS L x Z n Z+ x ny Z+

DS L x Z n, m Z+ { 0}

gm (gn (x)) = gn + m (x) = x + ( n + m) = x.

x tgt (x) x

x

DS L = ( M, (gt ) tT ) L = ( T, +)t T r T x M

gr

(gt

(x)) = x.


M 2 DS L 1 DS L 2 DS L 1 DS L 2


DS L 1 DS L 2 DS L 1 t1 T 1 v1 T 1

x1 M 1gv 1 (gt 1 (x1 )) = gv 1 + t 1 (x1 ) = x1

f (x1 ) = f (gv 1 + t 1 (x1 ))

= g ( v 1 + t 1 ) (f (x1 ))

= g ( v 1 ) ( t 1 ) (f (x1 ))

= g ( v 1 ) (g ( t 1 ) (f (x1 ))) .

f t2 T 2 x2 M 2 t1 T 1 x1 M 1t2 = (t1 ) x2 = f (x1 ) t2 T 2 v2 = (v1 ) T 2

x2 M 2gv 2 (gt 2 (x2 )) = g ( v 1 ) (g ( t 1 ) (f (x1 ))) = f (x1 ) = x2 ,


96/169

DS L 2 f 1 : M 2 M 1 1 DS L 2 DS L 1

DS L = ( M, (gt ) tT ) L = ( T, +) DS Lt T r T t T x M

r T DS L

tx z y

r y x z

DS L = ( M, (gt ) tT ) L = ( T, +) t T x, y M

gt (x) = gt (y);

r T

x = gr (gt (x)) = gr (gt (y)) = y;

gt DS L

DS L = ( M, (gt ) tT ) L = ( T, +)DS L t T x M

x /gt (M ),

gt (M ) gt

r T gr (gt (M )) = M,

gr (x) M.

y gt (M )

gr (y) = gr (x).


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gr

x = y

x gt (M ),

DS L DS L

DS L L DS LDS L

DS L = ( M, (gt )tT )L = ( T, +) t, r T x M

gr (gt (x)) = x,

gr = ( gt ) 1.

DS L = ( M, (gt ) tT ) L = ( T, +) DS L

t T gt

(gt

) 1

r, t T x M gr = ( gt ) 1 x M

gr (x) = ( gt ) 1 (x);

DS L gt y M y = gt (x)

gr (gt (y)) = ( gt ) 1 (gt (y)) = y,

r, t T x M gr = ( gt ) 1

DS L = ( M, (gt ) tT ) L = ( T, +)t T r T

gr = ( gt ) 1.

DS L = ( M, (gt ) tT ) L = ( T, +) DS L

t T r T x M r gr = ( gt ) 1


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DS L = ( M, (gt )tT ) L = ( T, +)t T r T x M

gt (gr (x)) = x.

DS L = ( M, (gt ) tT ) L = ( T, +) DS Lt T r T gr = ( gt ) 1

x M gt (gr (x)) = gt (( gt ) 1 (x)) = x.

DS L = ( M, (gt )tT ) L = ( T ; +)L

DS L 1 = ( M 1, (gt 1 )t 1T 1 ) L1 = ( T 1, +)DS L 2 = ( M 2, (gt 2 )t 2T 2 ) L2 = ( T 2,) f : M 1 M 2



DS L 1 DS L 2 DS L 1 L1 L2L2 DS L 1

f 1

: M 2 M 1 1

DS L 2 DS L 1

DS L = ( M, (gt ) tT ) L = ( T, +) 0 DS Lt T t T x M

g t (gt (x)) = g t + t (x) = g0 (x) = x;

DS L


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DS L L DS L

DS L L DS L

M = {x i }2i =1 L = ( Z+ , +)

n Z+ x i M

n gn (x i ) = x i ,

n gn (x i ) = x j ,

i = j

L = ( Z+ , +) 0

DS L = ( M, (gn )n Z + ) L M (gn )n Z +

M Z+ x i M

g0 (x i ) = x i

x i M n, m Z+

n m n + m

gn + m (x i ) = x i = gm (x i ) = gn (gm (x i )) ,

n m n + m

gn + m (x i ) = x j = gm (x i ) = gn (gm (x i )) ,

n m n + m

gn + m (x i ) = x j = gn (x i ) = gn (gm (x i )) ,

n m n + m

gn + m (x i ) = x i = gm (x j ) = gn (gm (x i )) .

DS L n Z+ m Z+ x i M

n m n + m

gm + n (x i ) = gm (gn (x i )) = x i


100/169

n m n + m

gm + n (x i ) = gm (gn (x i )) = x i .

DS L Z+

DS L = ( M, (gt )tT )L = ( T, +) (gt )tT DS L

DS L = ( M, (gt ) tT ) L = ( T, +) 0g : T M M (gt ) tT t T

r T

g(0) = g0 = ( gt ) 1 gt = gr gt = gr + t = g(r + t),

g(0) = g0 = gt (gt ) 1 = gt gr = gt + r = g(t + r );

g

0 = g 1 (g(0)) = g 1 (g(r + t)) = r + t

0 = g 1

(g(0)) = g 1

(g(t + r )) = t + r,

r t L r L DS L

DS LDS L

(f x )xX Y X f : X Y


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DS L = ( M, (gt )tT )

L = ( T, +) (gt)tT DS L

DS L = ( M, (gt ) tT ) L = ( T, +) 0g : T M M (gt ) tT g t, v T

g(t) = gt = gv = g(v) t = v.

t, v T r, u T

gr = ( gt ) 1

gu = ( gv ) 1

gr + t = gr gt = ( gt ) 1 gt = g0

= ( gt ) 1 gv = gr gv = gr + v

= ( gv ) 1 gv = gu gv = gu + v

= ( gv ) 1 gt = gu gt = gu + t

= gt (gt ) 1 = gv (gt ) 1 = gv gr = gv + r

= gv (gv ) 1 = gv gu = gv + u

= gt

(gv

) 1

= gt

gu

= gt + u

.i {t, v } j {r, u } t v r u i + j j + i0

t + ( r + v) = t + 0 = t = v = 0 + v = ( t + r ) + v,

+ i + j j + i 0 s T x M

gs (x) = g0 (x) = x s = 0 ,

s x M


102/169

DS L = ( M, (gt )tT )DS L

DS L L = ( T, +)DS L


103/169

DS L = ( M, (gt ) tT ) L = ( T, +)

t T gt x, z M

gt(x) = g

t(z ) = y x = z.

w, v T

gw (y) = x

gv (y) = z,

gw (gt (x)) = gw (gt (z )) = x

gv (gt (z )) = gv (gt (x)) = z

x = gw (gt (z )) = gw (gt (gv (gt (x))))

z = gv (gt (x)) = gv (gt (gw (gt (x)))) .

x = z

gw (gt (gv (gt (x)))) = gv (gt (gw (gt (x))))

+ T

t T gt x M y M

gt (x) = y

w T

gw (y) = x.

x M w T

gw (gt (x)) = x

gt (gw (x)) = x,

x M z = gw (x) M gt

DS L

T Z 0 M


104/169

t T x Z

gt (x) = t(x).

L = ( T, +)

T

0

DS L = ( Z, (gt ) tT ) L = ( T, +)

Z

(gt ) tT Z T

x Z t, v T

g0 (x) = 0( x) = x

gt v (x) = ( t v)( x) = t(v(x)) = gt (gv (x));

DS L t T t 1 T x Z

gt 1

(gt (x)) = t 1 (t(x)) = x,

DS L t T gt

tL = ( T, +) r T t + r + t = t


105/169

DS L = ( M, (gt ) tT ) L = ( T, +) Lt T r T t + r + t = t t, r T

x M gt (gr (gt (x))) = gt + r + t (x) = gt (x).

DS L t T r T

gr (gt (x)) = x

x M

gt (gr (gt (x))) = gt (x),

DS L

DS L = ( M, (gt ) tT ) L = ( T, +) Lt T r T t + r + t = t t, r T x M

gt (gr (gt (x))) = gt + r + t (x) = gt (x).

t T y M x M r T

y = gt (x);

gt (gr (y)) = gt (gr (gt (x))) = gt + r + t (x) = gt (x) = y;

DS L

DS L LDS L DS L DS L

DS L

DS L LDS L

DS L


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DS L = ( M, (gt )tT ) L = ( T, +)x, y M t, r T

gt (x) = gr (y)

v T gv (x) = y gv (y) = x.

DS L = ( M, (gt ) tT ) L = ( T, +) x, y M x M orb(x) y M orb(x) orb(y) orb(y) orb(x)

orb(x) orb(y) = x /orb(y) v T gv (y) = x v T gv (x) = y

z M z orb(x) z orb(y) t, r T gt (x) = z = gr (y) DS L

DS L L DS LDS L DS L

DS L = ( M, (gt )tT ) L = ( T, +) 0t T x M r T { 0}

gr (gt (x)) = x.

r = 0t = 0 x = x

r T { 0}


107/169

DS L = ( M, (gt )tT ) L = ( T, +)0 x M t T { 0}

r T { 0}gr (gt (x)) = x.

DS L = ( M, (gt ) tT ) L = ( T, +) 0 DS Lr T { 0} x M t T

r T { 0} x M t T { 0}r T { 0}

x M t T { 0}t = 0 t = 0 z M t T

gr (gt (x)) = gr (g0 (x)) = gr (x) = x.

gr (gr (x)) = gr (x) = x,

t = r T { 0}t T { 0}

DS L = ( M, (gt

) tT ) L = ( T, +) 0DS L x M t T r T { 0}

gr (gt (x)) = x;

t = 0 x M r T { 0}

gr (g0 (x)) = gr (x) = x.

DS L

DS L = ( M, (gt )tT ) L = ( T, +)x M

P (x) F (x) = .

DS L = ( M, (gt ) tT ) L = ( T, +) 0DS L x M t T { 0} v T { 0}gr (gt (x)) = x


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x, y M t T { 0} v T { 0} y F t (x) y P r (x)x, y M y F (x) y P (x) F (x) P (x) =

x M F (x) P (x) = T S L

DS L = ( M, (gt ) tT ) L = ( T, +) 0 x M DS LP (x) F (x) = F (x)

P (x) = F (x) DS L


109/169

7

L = ( T, +) T S L

T S L = ( I, ( t )tT )

I = T

t T i I t (i) = t + i.


110/169

I I

T S L L L

T S L = ( I, ( t ) tT ) L = ( T, +) 0

I

( t ) tT I T

i I t, v T

0 (i) = 0 + i = 0

t + v (i) = ( t + v) + i = t + ( v + i) = t (v (i)) .

T S L L

L = ( T, +) t T i I i t t (i)

i i t t (i)

t i

T I T

I L I


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L1 T S L 1 L2T S L 2 L2 L1 T S L 2 T S L 1

L1 = ( T 1 , +) T S L 1 = ( I 1 , ( t 1 ) t 1 T 1 ) L2 = ( T 2 ,)T S L 2 = ( I 2 , ( t 2 ) t 2 T 2 ) : T 2 T 1 L2 L1

t 2 T 2 i2 I 2 = T 2

( t 2 (i2 )) = (t2 i2 )

= (t2 ) + (i2 )

= ( t 2 ) ((i2 )) .

L1 T S L 1 L2T S L 2 L1 L2 T S L 1 T S L 2

L1 T S L 1 L2 T S L 2 L1 L2 T S L 2 T S L 1

T S L 2 T S L 1 T S L 2 T S L 1

DS L

L

DS LDS L L

DS L

DS L = ( M, (gt )tT ) L = ( T, +)DS L T ADS L

T ADS L = ( H, )

H = h : h = gt t T

DS L = ( M, (gt )tT )L = ( T, +) 0 g0

DS L = ( M, (gt ) tT ) L = ( T, +) 0 T ADS L = ( H, )DS L


112/169

gt , gv H gt gv = gt + v H

gt , gv , gw H

gt

(gv

gw

) = gt

gv + w

= gt +( v + w )

= g( t + v )+ w

= gt + v

gw

= ( gt

gv

) gw

g0 H h = gt H

g0 gt = g0+ t = gt = gt +0 = gt g0 .

T ADS L g0 T ADS L

L DS LT ADS L

DS L = ( M, (gt )tT ) L = ( T, +)(gt )tT L

T ADS L = ( H, ) DS L

DS L = ( M, (gt ) tT ) L = ( T, +) 0 T ADS L = ( H, )DS L g : T H (gt ) tT

g

g(0) = g0

;

g t, v T

g(t + v) = gt + v = gt gv = g(t) g(v).

g h H h = gt t

g(t) = gt ;

g L T AT S L

( t )tT

L

L = ( T, +) 0 T S L = ( I, ( t ) tT ) L T AT S L = ( H, )T S L : T H ( t ) tT

L T AT S L


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t, v T

t = v

t + 0 = v + 0

t

(0) = v

(0) t = v

(t) = (v);

L T AT S L L T AT S L

DS LT S T A DS L

L T ADS L

T S L L T AT S L T S L T S T A T S LT AT S L T AT S L L T S T A T S L

T S L

LT S L

L = ( T, +) t T t

T T t


114/169

T S L = ( I, ( t )tT ) L = ( T, +)

i I = T orb(i) i

T S L = ( I, ( t ) tT ) L = ( T, +)i I = T

orb(i) = j I : t T, j = t (i)

= { j I : t T, j = t + i} ,

i

T S L = ( I, ( t )tT ) L = ( T, +)0 orb(0) = I

T S L = ( I, ( t ) tT ) L = ( T, +) 0 t T

t (0) = t + 0 = t,

orb(0) = i I : t T, t (0) = i = T = I.

xDS L L DS x L L orb(x)

DS L orb(x)

T S L LDS 0L

T S L = ( I, ( t )tT ) L = ( T, +)0 T S L = T S 0L = ( orb(0) , ( t |orb (0) )tT )

T S L = ( I, ( t ) tT ) L = ( T, +) 0 I =orb(0) t T t |orb (0) = t ( t |orb (0) ) tT = ( t ) tT T S L = T S 0L =(orb(0) , ( t |orb (0) ) tT )


115/169

T S L = ( I, ( t )tT ) L = ( T, +)0 i I i i = 0

T S = ( I, ( t ) tT ) L = ( T, +) 0 i I i = 0

i = i + 0 = i (0);

t = i T { 0} j = 0 I t ( j ) = i ii I i i = 0

T S L L = ( T, +) 0i I 0 orb(i) i

T S L = ( I, ( t ) tT ) L = ( T, 0) 0 i T

0 orb(i) t T t (i) = 0

t T t + i = 0 ,

t i

T S L L = ( T, +) 0 T S LL

T S L = ( I, ( t ) tT ) 0 T S L0 i I { 0} t T

t (i) = 0

0 /orb(i)

i

T S L L = ( T, +) 0 0


116/169

T S L = ( I, ( t ) tT ) 0 t T { 0}

t (0) = t + 0 = t = 0;

0

T S L L = ( T, +) 0i I i orb(i) = I

T S L = ( I, ( t ) tT ) L = ( T, +) 0 i T

i t T = I

t(i) = t + i = 0

j I

j = j (0) = j (t + i) = j + ( t + i) = ( j + t) + i = j + t (i) orb(i);

orb(i) = I

orb(i) = I 0 orb(i)

i

T S L LL


117/169

T S = ( I, ( t ) tT ) L = ( T, +) 0

T S L i I t T r T

r + ( t + i) = i.

i = 0 t T r T

r + t = 0 ,

t T = I

L t T r T

r + t = 0;

j I

r(

t( j )) = r + t + j = 0 + j = j.

T S L

T S L

T S L L

T S L = ( I, ( t )tT ) L = ( T, +)i I orb(i) = I

T S L = ( I, ( t ) tT ) L = ( T, +) 0 T S Li I = T

i I orb(i) = I

xz y

y x z


118/169

T S L = ( I, ( t ) tT ) L = ( T, +) T S L

(Z+ , +)

L = ( T, +) t T u, v T t + u = t + v u = v

L t T u, v T u + t = v + t u = v

T S L L L

T S L = ( I, ( t ) tT ) L = ( T, +) T S Lt T i, j T = I

t (i) = t ( j ) i = j

t + i = t + j i = j,

L

T S L LL

T S L = ( I, ( t ) tT ) L = ( T, +) 0

T S L t T i I j I

t ( j ) = t + j = i;


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i = 0 t T j I = T

t + j = 0 .

L

t T i I = T

t + i = 0;

j I t (i + j ) = t + ( i + j ) = ( t + i) + j = 0 + j = j.

T S L

T S L = ( I, ( t ) tT ) L = ( T, +) 0

T S L i I j T = I

j + i = 0

k T = I

k + j = 0 .

k + ( j + i) = ( k + j ) + i

k + 0 = 0 + i

k = i.

j i T S L


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T S L = ( I, ( t ) tT ) L = ( T, +) 0 T S LT S L

i T j, k T

j + i = 0

i + k = 0

j = j + 0 = j + ( i + k) = ( j + i) + k = 0 + k = k.

L T S L


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DS L = ( M, (gt )tT ) L = ( T, +) 0x M F (x)

xx P (x)

xx

x

DS L = ( M, (gt )tT ) L = ( T, +) 0x M x (x)

(x)def = y M : y = g0(x) .

(Z+ )


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DS L = ( M, (gt )tT ) L = ( T, +)x M

(x) = {x} .

DS L = ( M, (gt ) tT ) L = ( T, +) 0 x M

(x) = y M : y = g0 (x) = {x} .

x

ii

ii

T S L = ( I, ( t )tT ) L = ( T, +)0 t T { 0} = I { 0}

F t (0) = ( t).

T S L = ( I, ( t ) tT ) L = ( T, +) 0 t T { 0} = I { 0}

F t (0) = j I : j = t (0) = { j I : j = t + 0 } = {t} = ( t).


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DS L 1 = ( M 1, (gt 1 )t 1T 1 ) DS L 2 = ( M 2, (gt 2 )t 2T 2 )L1 = ( T 1, +) L2 = ( T 2,) f : M 1 M 2

DS L 1 DS L 2 x1, y1 M 1

y1 F (x1) f (y1) F (f (x1))

y1 P (x1) f (y1) P (f (x1))

y1 (x1) f (y1) (f (x1))

DS L 1 = ( M 1 , (gt 1 ) t 1 T 1 ) L1 = ( T 1 , +) 0 DS L 2 =(M 2 , (gt 2 ) t 2 T 2 ) L2 = ( T 2 ,) : T 1 T 2L1 L2 f : M 1 M 2 DS L 1 DS L 2

y1 F (x1 ) t1 T 1 { 0}

gt 1 (x1 ) = y1 ,

(t1 ) T 2

(t1 ) = (0)

f (y1 ) = f (gt 1 (x)) = g ( t 1 ) (f (x1 ));

f (y1 ) F (f (x1 ))

y1 P (x1 ) x1 F (y1 )

f (x1 ) F (f (y1 )) f (y1 ) P (f (x1 )) y1 (x1 ) x1 = y1 f (x1 ) = f (y1 ) f (y1 ) P (f (x1 ))

f 1 1 DS L 2DS L 1


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DS L = ( M, (gt )tT ) L = ( T, +)x M

(x) F (x) =

P (x) (x) = ,

orb(x)

DS L = ( M, (gt) tT ) L = ( T, +) 0 x M

orb(x) x F (x)

(x) F (x) = .

orb(x) (x) F (x) = orb(x)x P (x)

P (x) (x) =

orb(x) P (x) (x) =

T S L = ( I, ( t )tT ) L = ( T, +)i I

(i) F (i) = ,

(i) P (i) = ,

P (i) F (i) =


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T S L

T S L = ( I, ( t ) tT ) L = ( T, +) i I T S L T S L

i I T S Li I

i I T = I

T

T

T S L = ( I, ( t )tT ) L = ( T, +)

i I P (i) (i) F (i) = I

j I

orb( j ) orb(i) orb(i) orb( j ).

T S L = ( I, ( t ) tT ) L = ( T, +) i I i j I j P (i) j ( i) j F (i)

orb(i) orb( j ) orb(i) = orb( j ) orb(i) orb( j ) orb( j ) orb(i)orb( j ) orb(i) j I orb(i) orb( j ) orb( j ) orb(i)

i orb( j ) j orb(i)i ( j ) i F ( j ) j (i) j F (i) j P (i) j ( i)

j F (i) j

T S L = ( I, ( t )tT ) L = ( T, +) i I

T S L


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T S L L i I i, j I orb(i) orb( j ) orb( j ) orb( j )

i I

T S L = ( I, ( t )tT ) L = ( T, +)i I T S L

T S L = ( I, ( t ) tT ) L = ( T, +) i I T S L


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8


128/169


129/169

S = ( D, (R i )iI ) D S

i {1,...,m,... } m 1 n R i n n 0S S

R i S S = ( D, (R i )iI )S = ( D , (R i )iI )

i I R i R i : D D S S

n (x1,...,x n ) S n R i R i (x1,...,x n )R i ((x1), ..., (xn ))


130/169

...,S (t0), S (t1), S (t2),......,S T(t2), S T(t1), S T(t0),... T


131/169

DS LDS L

DS L

t

tDS L = ( M, (gt )tT ) L = ( T, +) 0

t T t n x1,...,x n n > 1 x i

x i+1 = gt (x i ),

i = 1 ,...,n 1

T DS Ln (x1,...,x n ) DS L t

(T(xn ),..., T(x1))

DS L = ( M, (gt )tT ) L = ( T, +)DS L : M M


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x M t T

(gt ( (x))) = ( gt ) 1(x).

DS L 1 = ( M 1, (gt 1 )t 1T 1 ) DS L 2 = ( M 2, (gt 2 )t 2T 2 )L1 = ( T 1, +) L2 = ( T 2,) f : M 1 M 2



DS L 1 DS L 2 DS L 1 DS L 1DS L 2 : M 1 M 1 M 1

f : M 2 M 2 x2 M 2

(f 1 (x2 )) = f ( (f 1 (x2 ))) .

t 2 T 2 x2 M 2 t1 T 1 x1 M 1

t2 = (t1 ),

x2 = f (x1 );

gt 2 ((gt 2 ( (x2 )))) = g ( t 1 ) ( (g ( t 1 ) ((f (x1 )))))

= g ( t 1 ) ( (g ( t 1 ) (f ( (x1 )))))

= g ( t 1 ) ( (f (gt 1 ( (x1 )))))

= g ( t 1 ) (f ( (gt 1 ( (x1 )))))

= f (gt 1 ( (gt 1 ( (x1 )))))

= f (x1 )

(gt 2 ( (x2 ))) = (g ( t 1 ) ((f (x1 ))))

= ( g ( t 1 ) ) 1 (( f (x1 ))) .

f t2 T 2 x2 M 2 t1 T 1 x1 M 1 M 2

DS L 2 f

x M (x) M x


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DS L = ( M, (gt )tT )L = ( T, +) 0 : M M

DS L = ( M, (gt ) tT ) L = ( T, +) : M M M x M

( (x)) = (g0 ( (x))) = ( g0 ) 1 (x) = x.

DS L = ( M, (gt )tT ) L = ( T, +) : M M x M

t T gt ( (gt ( (x)))) = x.

gt

DS L

DS L = ( M, (gt ) tT ) L = ( T, +) x M t T DS L : M M

x M gt ( (gt ( (x)))) = x.

(gt ( (gt ( (x))))) = (x)

(x) x

(gt ( (gt ( ( (x)))))) = ( (x)) .

t = 0( (x)) = x

M t T gt


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t T gt gt x, y M x = y

(gt (x)) = (gt (y))

( (gt (x))) = ( (gt (y)))

gt

(x) = gt

(y);

t T gt gt x M y M

(gt (y)) = x

( (gt (y))) = (x)

gt (y) = (x).

z M x M (x) = z x x z M y M

gt (y) = z.

DS L = ( M, (gt

) tT ) L = ( T, +)

DS L : M M t T x M

(gt ( (x))) = ( gt ) 1 (x)

gt ( (gt ( (x)))) = gt (( gt ) 1 (x)) = x;

DS L

DS L : M M t T x M

gt ( (gt ( (x)))) = x.

DS Lgt (gt ) 1 t T

x M (gt ( (x))) = ( gt ) 1 (gt ( (gt ( (x))))) = ( gt ) 1 (x).

DS L

DS L = ( M, (gt )tT ) L = ( T, +)M

DS L = ( M, (gt ) tT ) L = ( T, +) : M M


135/169

M x M t T

(gt ( (x))) = ( gt ) 1 (x)

gt ( (gt ( (x)))) = gt (( gt ) 1 (x))

= x;

M

: M M M

x = gt ( (gt ( (x)))) .

DS Lt T gt

(gt ) 1 (x) = ( gt ) 1 (gt ( (gt ( (x)))))

. = (gt ( (x))) .

M

L = ( T, ) Z Z x Z

gt (x) = t(x).

L = ( T, )

T

t T t 1 T

t t 1 = = t 1 t ;

DS L = ( Z, (gt ) tT ) L = ( T, )

Z

(gt ) tT Z T

x Z t, w T

g

(x) = (x) = xgt w (x) = ( t w)( x) = t(w(x)) = gt (gw (x));


136/169

DS L T

DS L x Z t T : Z Z

w T

= w

x M t T

gw (gt (gw (x))) = ( gt ) 1 (x)

gw (gw (gt (gw (x)))) = gw (( gt ) 1 (x)) gw

gt (gw (x)) = gw (( gt ) 1 (x)) gw

gt (gw (x)) = ( gw ) 1 (( gt ) 1 (x)) gw

gt (gw (x)) = ( gt gw ) 1 (x) t, w T w 1 t 1 = ( t w) 1

gt (gw (x)) = gw (gt (gw (gw (x))))

gt (gw (x)) = gw (( gt (x))) gw

gw (gt (gw (x))) = gw (gw (gt (x))) gw

(gt ) 1 (x) = gw (gw (gt (x)))

(gt ) 1 (x) = gt (x) gw

Z

L = ( Z+ , +)n Z+ gn : Z Z x Z

gn (x) = n + x

: Z Z x Z

(x) = x.

DS L = ( Z, (gn )n Z + ) L = ( Z+ , +)

Z

(gn )n Z + Z Z+

n, m Z+ x Z

g0 (x) = 0 + x = x,

gn + m (x) = ( n + m ) + x = n + ( m + x) = gn (gm (x));


137/169

DS L x M n Z+

(gn ( (gn (x)))) = (gn ( (gn (x)))) = (n + ( (n + x))) = ( x) = x;

DS L x Z n Z+ m Z+ { 0}

gm (gn (x)) = m + ( n + x) = ( m + n) + x = x.

tt

DS L = ( M, (gt )tT )L = ( T, +) 0 : M M M

x, y M t T

y = gt (x) (x) = gt ( (y)) .


138/169

DS L = ( M, (gt ) tT ) L = ( T, +) 0: M M M t T x, y M

y = gt (x) (gt (x)) = (y)

(gt ( ( (x)))) = (y)

(gt ) 1 ( (x)) = (y)

(x) = gt ( (y))

t t

DS L = ( M, (gt )tT )L = ( T, +) 0 t T x i M i = 1 ,...,j,...,n (x1,...,x j )(x j ,...,x n ) t (x1,...,x j ,...,x n ) t

DS L = ( M, (gt ) tT ) L = ( T, +) 0t T x i M i = 1 ,...,j,...,n (x1 ,...,x j ) (x j ,...,x n ) t

i = 1 ,...,j 1 k = j,...,n 1l = 1 ,...,j 1,j,...,n 1 (x1 ,...,x n +1 ) t (x1 ,...,x n +1 )

t i = 1 ,...,n 1 i = 1 ,...,j 1k = j,...,n 1 (x1 ,...,x j ) (x j ,...,x n ) t

tt

DS L = ( M, (gt )tT )L = ( T, +) 0 : M M M

x i M i = 1 ,...,n t T (x1,...,x n ) t( (xn ),..., (x1)) t

DS L = ( M, (gt

) tT ) L = ( T, +) 0: M M M t T x i M i = 1 ,...,n

n

n = 2 (x1 , x 2 ) t x2 = gt (x1 ) (x1 ) = gt ( (x2 )) ( (x2 ), (x1 )) t

n = m (x1 ,...,x m ) t ( (xm ),..., (x1 )) t

n = m + 1 (x1 ,...,x m +1 ) t (x1 ,...,x m ) (xm , x m +1 ) t( (xm +1 ), (xm )) t ( (xm ),..., (x1 ))

t ( (xm +1 ),..., (x1 )) t


139/169

DS L = ( M, (gt )tT ) L = ( T, +): M M x i M i = 1 ,...,n

t T (x1,...,x n ) t ( (xn ), ..., (x1)) t : M M M

DS L = ( M, (gt ) tT ) L = ( T, +) 0 : M M x i M i = 1 ,...,n t T (x1 ,...,x 2 ) t

( (xn ),..., (x1 )) t x, y M (x, y ) t

n = 2 x, y M t T

y = gt (x).

(x) = gt ( (y)) ,

(x) = gt ( (gt (x))) .

x (x)

x = ( (x)) = gt ( (gt ( (x)))) ,

M

DS L = ( M, (gt )tT ) L = ( T, +): M M x i

M i = 1 ,...,n t T (x1,...,x n ) t ( (xn ), ..., (x1))

t

DS L = ( M, (gt ) tT ) L = ( DS L ) DS LM

: M M M x i M i = 1 ,...,n t T n (x1 ,...,x n ) t

( (xn ),..., (x1 )) t


140/169

DS L = ( M, (gt ) tT ) L = ( T, +) f : M M

DS L : T T f DS L DS L

DS L L = ( T, +)DS L

DS L = ( M, (gt ) tT ) L = ( T, +) 0 ( DS L )DS L

(DS L ) (( DS L ), )

f 1 , f 2 ( DS L ) DS L1 : T T 2 : T T f 1 1 DS L DS L f 2

2 DS L DS L

1 2 : T T L

x M t T

f 1 f 2 (gt (x)) = f 1 (f 2 (gt (x)))

= f 1 (g 2 ( t ) (f 2 (x)))

= g 1 ( 2 ( t )) (f 1 (f 2 (x)))

= g 1 2 ( t ) (f 1 f 2 (x)) ,

f 1 f 2 1 2 DS L DS L

1 2 1 2 f 1 f 2DS L f 1 f 2 ( DS L ) (( DS L ), )


141/169

M T

DS L DS L x M t T

(gt (x)) = gt (x) = g ( t ) ((x));

DS L (DS L )f DS L x M t T

f (gt (x)) = f ((gt (x))) = f (gt (x)) = (f (gt (x))) = f (gt (x)) ,

( DS L )

f (DS L )) f DS L DS L

f 1 : M M 1 : T T

f 1 1 DS L DS L x M t T

f 1 (gt (x)) = f 1 (g ( 1 ( t )) (f (f 1 (x))))

= f 1 (f (g 1 ( t ) (f 1 (x))))

= g 1

(f 1 (x))

x M t T

f 1 f (gt (x)) = f 1 (g ( t ) (f (x))) = g 1 ( ( t )) (f 1 (f (x))) =

= e(gt (x)) =

= g ( 1 ( t )) (f (f 1 (x))) = f (g

1 ( t ) (f 1 (x))) = f f 1 (gt (x));

f 1 DS L (( DS L ), )


142/169

DS L = ( M, (gt )tT )L = ( T, +) g : T M M (gt )tT

: M M DS L : T T t T

g((t)) = ( g(t)) 1.

DS L = ( M, (gt ) tT ) L = ( T, +) g : T M M

(gt ) tT : M M M

x M t T

(gt (x)) = (gt ( ( (x)))) = ( gt ) 1 ( (x)) .

DS L : T T t T x M

(gt (x)) = g ( t ) ( (x)) = ( gt ) 1 ( (x)) .

(x) x

g ( t ) (x) = ( gt ) 1 (x),

t T t T x M (x) x

DS L = ( M, (gt )tT )L = ( T, +) g : T M M (gt ) tT : T T

t T DS L

DS L = ( M, (gt ) tT ) L = ( T, +) g : T M M

(gt ) tT : T T t T t T r T

r = (t ),

gr = g(r ) = g((t)) = ( g(t )) 1 = ( gt ) 1 ;

DS L


143/169

DS L = ( M, (gt )tT ) L = ( T, +) g : T M M

(gt )tT L t T

DS L = ( M, (gt ) tT ) L = ( T, +) g : T M M

(gt ) tT : T T T t T

(t) = t ;

t T g((t)) = g( t) = g t = ( gt ) 1 = ( g(t )) 1 .

L = ( T, +) : T T t T L L

L = ( T, +) 0 : T T t T t T

((t)) = ( t) = t.

t, v T

t + v = ((t + v)) = ((t) + (v)) = ( t + ( v)) = ( (v)) + ( ( t )) = v + t,

L LL

DS L = ( M, (gt )tT ) L = ( T, +) L

: M M DS L


144/169

DS L = ( M, (gt ) tT ) L = ( T, +) g : T M M

(gt ) tT L : T T t T

M DS L

L = ( T, +) DS L =(M, (gt )tT ) DS L

M DS L


145/169

DS L = ( M, (gt ) tT ) L = ( T, +)g : T M M : T T

DS L = ( M, (gt ) tT ) L = ( T, +) g : T M M

: T T : T T gt T

(t ) = g 1

(g((t ))) = g 1

(( g(t)) 1

) = g 1

(g( (t))) = (t),

DS L = ( M, (gt )tT )L = ( T, +) (gt ) tT : M M DS L L

DS L = ( M, (gt ) tT ) L = ( T, +)(gt ) tT L

M DS L : M M DS L DS L

: T T

t T L L


146/169

T S L = ( I, ( t )tT ) L: I I L

T S L = ( I, ( t ) tT ) L = ( T, +) : I I ( t ) tT L T AT S L T S L

T S L = ( I, ( t ) tT ) L = ( T, +) 0 : I I i I

(i) = i;

i I t T = I

(t i) + ( (t i)) = 0 i + ( (t i)) = t

(t i) = i + ( t).

( t ( (i))) = ( t ( i)) = (t i) = i + ( t) = t + i = ( t ) 1 (t ).

T S L T S L


147/169

L = ( T, +) T S L = ( I, ( t )tT )( t )tT t T

T S L

DS L

T S L

DS LDS L L


148/169


149/169


150/169

DS L = ( M, (gt )tT ) L = ( T, +)DS L L = ( T , + |T ) L DS L |L

DS L |L = ( M, (gt )tT )

DS L = ( M, (gt ) tT ) L = ( T, +) 0 L = ( T , + |T )L DS L | L = ( M, (g

t ) tT ) DS L L DS L | L =(M, (gt ) tT ) L

L = ( T, + , ) 0L+ = ( T + , + |T + , | T + ) T + = {t T : 0 t} + |T +

+ T + | T + L = ( T, + , ) L = ( T , + |T , | T )

T = {t T : t 0}

L = ( T, + , ) 0 L+ = ( T + , + |T + , | T + ) L =(T , + |T , | T ) L+

L T + LL+ L T +

+ |T + t, v T + { 0}


151/169

t + |T + v T T + T

t + |T + v t t + |T + v v,

t + v t t + v v,

( t + t) + v t + t t + ( v + ( v)) v + ( v),v 0 t 0,

v T t T t = 0 v = 0T + | T + | T +

+

L T S LL T S L

T S LL = ( T, + , ) 0 L+ =

(T + , + |T + , | T + ) L = ( T , + |T , | T ) L=(T , + |T , | T ) L

: T T t T

|T + : T + T

(T + , + |T + ) (T , + |T ) L

(T +

, | T + ) (T

, | T )|T : T T +

(T , + |T ) (T + , + |T + ) L

(T , | T ) (T + , | T + )

L = ( T, + , ) 0 L+ = ( T + , + |T + , | T + )L = ( T , + |T , | T ) L = ( T , + |T , | T )

L : T T t T |T + : T + T T + L+

|T + (T + , | T + ) (T , | T )

t T + t T t T

t T +

t, v T +

t v

|T + (v) + t + |T + (t) |T + (v) + v + |T + (t)

|T + (v) |T + (t)|T + (t) |T + (v).


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|T + (T + , + |T + ) (T , + |T ) L

|T + (0) = 0 = 0;

L

L t, v T +

(t + |T + v) = (t + v) = (t + v) = v+( t ) = t +( v) = (t)+ (v) = (t)+ |T (v)

L t, v T + |T +

t + |T + v = v + |T + t

(t + |T + v) = (v + |T + t)

= (v + t)= (v + t)

= t + ( v)

= t + |T ( v)

= (t ) + |T (v).

DS L = ( M, (gt )tT )L = ( T, + , ) L+ = ( T + , + |T + , | T + )

L = ( T , + |T , | T ) L= ( T , + |T , | T )L : T T

t T |T + : T + T |T + : T T T + T

M |T + DS L |L + DS L |L

M |T DS L |L DS L |L +

DS L = ( M, (gt ) tT ) L =(T, + , ) L+ = ( T + , + |T + , | T + ) L = ( T , + |T , | T )L+ L L L = ( T , + |T , | T )

L DS L | L + = ( M, (gt ) tT + ) DS L | L =

(M, (gt

) tT ) DS L : T T t T |T + : T + T


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tt

i = t t


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http://philosophyfaculty.ucsd.edu/faculty/ccallender/

TimesOnticVoltage.doc

http://

philosophyfaculty.ucsd.edu/faculty/ccallender/FQX.pdf
http://philosophyfaculty.ucsd.edu/faculty/ccallender/Time'sOnticVoltage.dochttp://philosophyfaculty.ucsd.edu/faculty/ccallender/Time'sOnticVoltage.dochttp://philosophyfaculty.ucsd.edu/faculty/ccallender/FQX.pdfhttp://philosophyfaculty.ucsd.edu/faculty/ccallender/FQX.pdfhttp://philosophyfaculty.ucsd.edu/faculty/ccallender/FQX.pdfhttp://philosophyfaculty.ucsd.edu/faculty/ccallender/FQX.pdfhttp://philosophyfaculty.ucsd.edu/faculty/ccallender/Time'sOnticVoltage.dochttp://philosophyfaculty.ucsd.edu/faculty/ccallender/Time'sOnticVoltage.doc


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http://www.webalice.it/

marcogiunti/download/papers/GiuntiRedDS1finaleWEB.pdf

http://www.webalice.it/marcogiunti/download/

papers/DecompDynSys(web).pdf

http://www.

webalice.it/marcogiunti/download/papers/RARinDS(web).pdf

http://www.alophis.unica.it/files/Dynamical%20Systems%20on%20Monoids.pdf
http://www.webalice.it/marcogiunti/download/papers/GiuntiRedDS1finaleWEB.pdfhttp://www.webalice.it/marcogiunti/download/papers/GiuntiRedDS1finaleWEB.pdfhttp://www.webalice.it/marcogiunti/download/papers/DecompDynSys(web).pdfhttp://www.webalice.it/marcogiunti/download/papers/DecompDynSys(web).pdfhttp://www.webalice.it/marcogiunti/download/papers/RARinDS(web).pdfhttp://www.webalice.it/marcogiunti/download/papers/RARinDS(web).pdfhttp://www.alophis.unica.it/files/Dynamical%20Systems%20on%20Monoids.pdfhttp://www.alophis.unica.it/files/Dynamical%20Systems%20on%20Monoids.pdfhttp://www.webalice.it/marcogiunti/download/papers/RARinDS(web).pdfhttp://www.webalice.it/marcogiunti/download/papers/RARinDS(web).pdfhttp://www.webalice.it/marcogiunti/download/papers/DecompDynSys(web).pdfhttp://www.webalice.it/marcogiunti/download/papers/DecompDynSys(web).pdfhttp://www.webalice.it/marcogiunti/download/papers/GiuntiRedDS1finaleWEB.pdfhttp://www.webalice.it/marcogiunti/download/papers/GiuntiRedDS1finaleWEB.pdf


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http://www.alophis.unica.it/files/

Reversible%20Dynamics%20and%20the%%20Directionality%20of%20Time.pdf
http://www.alophis.unica.it/files/Reversible%20Dynamics%20and%20the%%20%20Directionality%20of%20Time.pdfhttp://www.alophis.unica.it/files/Reversible%20Dynamics%20and%20the%%20%20Directionality%20of%20Time.pdfhttp://www.alophis.unica.it/files/Reversible%20Dynamics%20and%20the%%20%20Directionality%20of%20Time.pdfhttp://www.alophis.unica.it/files/Reversible%20Dynamics%20and%20the%%20%20Directionality%20of%20Time.pdf


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http://philsci-archive.pitt.edu/archive/00004829/01/flow-for-archive.pdf
http://philsci-archive.pitt.edu/archive/00004829/01/flow-for-archive.pdfhttp://philsci-archive.pitt.edu/archive/00004829/01/flow-for-archive.pdf


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http://web.mit.edu/bskow/www/research/rate-of-passage.pdf
http://web.mit.edu/bskow/www/research/rate-of-passage.pdfhttp://web.mit.edu/bskow/www/research/rate-of-passage.pdf


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Documents

Temporal Becoming and the algebra of time