A N D R Z E J S-LUZA LEC

P R O J E C T I O N S Y S T E M S D E S C R I B E D I N

L A T T I C E T H E O R Y

1. I N T R O D U C T I O N

This paper analyses projections in projective and affine space, which are

considered as lattices (see [1]). The most impor tant methods of projection

used in geometry are those which enable us to reconstruct a manifold from

its projections and traces. To reconstruct a manifold it usually does not

suffice to know the projection from one centre on to one plane; thus every

method of projection is characterized by a system of several projections.

Such projection systems will be analysed in this paper.

2. n - S P A C E S AS L A T T I C E S

By (L,¢, u , n , 0, l ) we denote a lattice, where &a is a set, w, r~ are

operations ~2__~ ~, 0 is the zero and 1 is the unit of the lattice. By an

n-dimensional metric lattice, or projective space, we shall mean a structure

(~Ca, w, c~, 0, 1, d im) , where (L~ a, ~ , n , 0, 1) is a lattice and dim: ~ - ~

{ - 1, 0, 1 . . . . , n} is a function of dimension; for properties of dim see [1].

By an n-dimensional affine space we shall mean the structure (Z~a, u , c~, 0, 1,

h, d im) , where ( ~ , u , n , 0, 1, d im) is n-dimensional projective space,

h ~ L f ' , and d i m ( h ) = n - 1 . The element h will be called an ( n - 1 ) -

hyperplane at infinity. The set J = {x ~ ~ : x c h} will be known as the set

of manifolds at infinity. ~ = {x ~ ~e : ~ x c h} will be called the set of proper manifolds.

Manifolds al, a 2 . . . . , ae in a projective space are said to be independent if

dim a i = dim(a/) + p - 1. i i=1

It is not diff icult to prove that in a projective space a n (sl u s2) = 0 and

sl n s 2 = 0 i m p l y that a = 02=1 (a u si).

We now come to the formulat ion and proof of two lemmas which consti-

tute a p roof of Theorem 3 given below. The author is indebted to the referee for providing shortened proofs of these lemmas.

Assume sl, $2 . . . . , s, are independent manifolds at infinity in an affine

space, where u/> 2, and a is a proper manifold. Let q be an integer, l < ~ q < u .

Geometriae Dedicata 18 (1985) 35-41. 0046-5755/85.15. © 1985 by D. Reidel Publishing Company.

36 A N D R Z E J S-I:: U Z A L E C

1. L E M M A . I f a n U i~i s i = fg for some I ~_ {1, 2 . . . . . u} with

I I I - - - q + 1, then a = N (a w Uj~s sj), the intersection taken over all J ~_ 1

with I JI = q.

2. L E M M A . I f a n U i e i si # f~3for all I ~ {1, 2 . . . . . u} with [ I I = q + 1,

then dim(a) >~ (u - 1)/q.

Proof of Lemma 1. By i n d u c t i o n on q.

Case q = 1. The p r o o f is i m m e d i a t e f rom the m o d u l a r law a n d the inde-

p e n d e n c e of a, s l , s2 ( i f I = {1, 2}).

Case q > 1. A s s u m e the s t a t e m e n t h o l d s for q - - 1 . Le t I -- {1, 2 . . . . .

q + 1}. W e use the n o t a t i o n ~j to i nd ica t e t ha t sj is d r o p p e d f rom the

sequence .

(a u sl u " " ~ Sq) n ~ (a u S 1 U " ' " g Sj g " ' ' U S q + l )

l<~j<~q

= N [ ( a u Sl u ' ' ' u Sq) n

{(a u s, u . . . u ~ u . . . u sq) u s q ÷ J ]

(using the m o d u l a r law)

= n E ( a u s l u u s i u ' " U S q ) U l<~j~q

{(a u s 1 u " " u sq) c~ Sq+1} ]

(since (a u s l u "-" u Sq) n Sq+ 1 = ~ )

= n (a U Sl kA " '" U SJ U " ' " U S q ) = a l<~j<~q

(by induct ion) .

Proof of Lemma 2. A g a i n we use i n d u c t i o n on q.

Case q = 1. W e n o w use i n d u c t i o n on u. The case u = 2 be ing tr ivial ,

a s s u m e tha t u > 2 a n d t ha t the s t a t e m e n t is p r o v e d for u - 1. If a n sj # ~Z~

for a l l j = 1, 2 . . . . , u - 1 then d im(a n Ul-<i -< , -1 st)~> u - 1, a n d we are

done . If, o n the o t h e r h a n d , a n sj = ~5 for s o m e j <~ u - 1, then

(1)

(2)

(3)

a~(s~s.)~ U s ,=ansj=~, l <~i<~u 1

a ~ (sj u s,) # ~ (by a s sumpt ion ) ,

d im(a n U sl)/> u - 3 (by i n d u c t i o n hypothes is ) . l <~i <~n-1

P R O J E C T I O N S Y S T E M S AS L A T T I C E S 37

F r o m (1), (2), and (3) together it follows tha t

d i m ( a c ~ 1.<i~.U s l ) > > ' u - 2 , / so d i m ( a ) ~ > u - 1 .

Case q > 1. Assume the s ta tement holds for q - 1. This case is p roved by

induc t ion on u, s tar t ing from u = q + 1.

(a) u -- q + 1. Since a is p rope r and intersects the hyperp lane at infinity,

u - - I dim(a) /> 1 -

q

(b) u > q + 1. The s ta tement holds for u - 1.

Assume first tha t a n (sl u . . . u su) = a n (sl u . . . u s ,_ l ) . F r o m the

as sumpt ions for L e m m a 2, we get in this case that

a n U s j ¢ ~ j ~ J

for all J _~ {1, 2, . . . , u - 1} with I J L = q. Hence we derive from the induc-

t ion a s sumpt ion tha t

u - - 2 dim(a) ~> - -

q - l '

and the la t ter is ~>(u - l) /q; hence we are done in this case.

N o w consider the case that a n (sl u . . . u s,) ¢ a ~ (Sl w . . . u s , - O . Then

dim(a n (sl u -." u s,)) >~ dim(a m (sl u - . - ~ s , _ l ) ) + 1.

The induct ion hypothes is (on u) implies tha t

u - 2 d i m ( a n ( s 1 ~ " " u s u _ l ) ) ~ > - - 1.

q

Thus

dim(a) ~> dim(a c~ (s 1 u - . . u s,)) + 1

u - 2 u - 1 ~ > - - + l > - -

q q

This comple tes the proofs of the lemmas.

C o m b i n i n g Lemmas 1 and 2 we get the fol lowing theo rem:

38 A N D R Z E J S-'lz U 7, A L E C

3. T H E O R E M . Le t q > 0, u / > q + 1. Assume sl, s2 . . . . , s u are independent

manifolds at infinity in an affine space. Then for every proper manifold a with dim(a) <~ (u - 2)/q we have

where I runs ove r all subse ts of { 1, 2 . . . . . u} such t ha t I I I = q.

3. P R O J E C T I O N

By a projection sys tem in a p ro j ec t ive space we u n d e r s t a n d a pa i r F = (s, r)

of man i fo ld s s (cal led the centre) a n d r (cal led the projection manifold).

F is ca l led semiregular if s u r = l , a n d regular if s u r = 1 and

s ~ r = O .

The m a n i f o l d (a u s) n r wil l be ca l led the projection of the e l emen t a

(wi th respec t to F = (s, r)); we d e n o t e this by a e. The m a p p i n g # : ~ - - ~ L,e

de f ined by #(a) = (a ~ s) r~ r is ca l led the p r o j e c t i o n wi th respec t to F. The

m a n i f o l d a c~ r wil l he ca l led the trace of the e l emen t a (wi th respec t to

F = (s, r)), d e n o t e d by a r .

4. T H E O R E M (see [2]). I f the projection sys tem F = (s, r) is semiregular, then a u s = a e U s.

If we have severa l p r o j e c t i o n sys tems F 1 = (sl , r 0 , F 2 = ( $ 2 , r 2 ) . . . . .

Fp = (se, re), then the p r o j e c t i o n of a m a n i f o l d a (wi th r e spec t to F~ =

(si, rl), i ~ { 1, 2 . . . . . p}) is d e n o t e d by a i a n d the t r ace by a~.

In affine space two p r o p e r m a n i f o l d s a a n d b are ca l led parallel ( n o t a t i o n :

aHb)i fO ~ a n h o b n h o r O ~ b n h c a n h.

5. D E F I N I T I O N . In affine space a p r o j e c t i o n wi th respec t to a p r o j e c t i o n

sys tem F = (s, r) wil l be ca l led a parallel projection if F is s emi regu la r , s a

m a n i f o l d a t inf ini ty, a n d r a p r o p e r man i fo ld .

4. R E C O N S T R U C T I O N OF M A N I F O L D S

A s s u m e tha t in a p ro j ec t ive space we have p p r o j e c t i o n sys tems F i = (sl, r~),

i = 1, 2 . . . . , p. If the p p r o j e c t i o n s a ~ (i = 1, 2 . . . . . p) of the m a n i f o l d a (with

respec t to F~) a re k n o w n , then we on ly k n o w tha t the m a n i f o l d a is i nc luded

in the m a n i f o l d s a i u sl (i = 1, 2 . . . . . p). W e say t ha t the m a n i f o l d a is

reconstructed f rom its p r o j e c t i o n a 1, a 2 . . . . . a P w h e n e v e r a = N/e= 1 ( ai W Sl).

P R O J E C T I O N SYSTEMS AS LATTICES 39

By Theo rem 3 and 4 we have:

6. T H E O R E M . Le t q > 0, u ~> q + 1. Assume t 1 . . . . . t, are independent

manifolds at infinity in an affine space, s 1 . . . . . se , with

p=(;) the different manifolds

s i = U t J , J e ~ - { 1 , 2 , . - . , u } , I J i = q . J ~ J i

Assume, moreover, that r 1, . . . , r e are proper manifolds. Then every proper

manifold a with dim(a)<~ ( u - 2)/q can be reconstructed f r o m its parallel

projections a 1, a 2 . . . . . a P with respect to the projection systems Fi = (si, ri).

Suppose tha t in an affine space we have p paral le l p ro jec t ion systems F~ =

(s~, r~), i = 1, 2, . . . , p. The fol lowing theorems hold :

I f a, b can be reconstructed f r o m their projections and are neither empty nor

points, then a~llb i f o r every i ~ {1, 2, . . . , p} implies al[b. I f a c~ s i = 0 and

b ~ si = O for every i e {1, 2, . . . , p}, then a[lb implies a*llb~for every i e {1, 2,

• . . ~ p } .

M o r e o v e r :

I f ai, bl are proper manifolds and are not points, then a[[b implies ail[b i for

every i E {1, 2 , . . . , p}.

The proofs of these theorems are left to the reader.

A C K N O W L E D G E M E N T

! would like to express my gra t i tude to the referee for his va luable advice.

R E F E R E N C E S

1. Birkhoff, G., 'Lattice Theory', Colloquium Publ., Vol. 25, Amer. Math. Soc., Prov., 1967. 2. Stu~:alec, A., 'Some Remarks on Projection Methods', Demons. Mathematica 2 (1984), 339-

353.

40 A N D R Z E J S4z U2;A LE C

(Received March 1, 1984; revised version, May 24, 1984)

Author's address:

Andrzej Stu2alec, Technical University of Cz~stochowa, 42-201 Cz¢stochowa, Poland