I n t h i s final s e c t i o n we consider s o m e i n t r i n s i c c o n d i t i o n s t h a t a n i z o p o s e t L m a y satisfy, w h i c h allow us to a s s u m e t h a t t h e i m a g e of g: O X ~ L in section 4 s h a h s a t i s f y C o n d i t i o n s L2, La, L5 a b o v e . T h e choice of i n t r i n s i c c o n d i t i o n s is i m p o s e d on us n a t u r a l l y
230 H. B. GRIFFITHS
because of our policy of working inductively 'downwards' from 1 through the co-atoms of L. Always, L is an izoposet with non-empty sets L , , L* of atoms and co-atoms. The first two conditions are
A~: For all co-atoms i, ft EL*, i / i :~ft then 2 n # e (2.L)* n (#.L)*.
A2: Given x E L . and 2 -( # ~( 1, a unique lub x V 2 EL is de/ined and x V 1 :~ 1.
(Thus xV~>~x, x V t > ~ i , whence x V i = i if x 6 t ; and if also x V 2 ' is defined and 2>~I ', then x V2 >~x V 2'). Because of the possibility of parallels, condition A 2 is not satisfied b y the lines and hyperplanes in the lattice of flats of Euclidean space, if hyperplanes are atoms and 1 is the e m p t y flat. The condition implies that atomic complements are co-atoms, by the next lemma; and the later ones (17.2, 17.3) are weak forms of the modular l a w - - n o t surprisingly since A 1 will hold whenever the dual L ~ of L is semi-modular (see 17.19).
17.1 L~MMA. I [ ~ 6 L * , vE(~.L)*, and x E L , , x r then x vvEL*.
Proo/. Since v C i ~ 1, then by A2, ~ = x V v is defined in L and ~ 4:1. Hence there exists # EL* such t h a t ~ ~<ft. Then b y the isotone properties of" N and V,
Now # fl i + 2 , otherwise x E # c 2, contrary to x f2; also by
A1,/A
n ~ e(~.L)*,
and since dis- tinct co-atoms of 2.L are not comparable, then v = # n ~. B u t again b y A1,/z N ~ E (#.L)*, so is a co-atom in #.L. However, ~ > v, otherwise x E a = v ~< 2, contrary to hypothesis. B u t 6/z.L, so ~ =ft. Therefore x V ~ EL*, as asserted.17.2 Lv, M~A. With ~ and x ~l, as above, (x V ~,) N 2 = ~.
Proo]. We have just seen t h a t a = x V v E L * . Since xE:r and xr then cr Hence N 2 6 (1.L)*, by A~. B u t ~ N 2 >~ 9 N i = 9 (since v el.L), so ~ N 2 = v since distinct co-atoms are not comparable. This completes the proof.
17.3 COROLL~Y.~6(xVr.L)*.
For, b y Lemma 17.2, v = (x V v) N t 6 (x V v.L)* by A~.
17.4 LEM~-~. I / x i s a n a t o m s u c h t h a t x r x E # , a n d 2 , f t 6 L * t h e n x v ( l n f t ) =ft.
Proo/. Since x E# and t N f t < f t , then the lub property of V implies t h a t y = Thus ye/~.L. Also, b y A~,
T H E H O M O L O G Y G R O U P S O F S O M E O R D E R E D S Y S T E M S 231
#~7~4
n/~ E (/~.L)*.B u t x E7, a n d x ~4 n/~, so y > 4 N # in/~.L. Therefore 7 : j u , as required.
F o r each a t o m x E L , , define L*(x) as in section 5 b y L*(x) = {/~ EL*Ix E~}.
Suppose 4EL*-L*(x); we prove t h e following lemma, whose i n t e r p r e t a t i o n in F l a t (pn) is familiar. (It is p a r t of Condition L 4 in section 10).
17.5 LEMMA. There is a bisection 4: L*(x) ~(4.L)*, given by r = ~ n4. (~r
Proo]. B y A1, r is a function. I t is onto, because each v E (2.L)* can be written v = (xV~) n 4 b y 17.2, a n d x V v E L ( x ) b y 17.1.
To see t h a t r is one-one, suppose r =r i.e. ju N 4 = # ' n 4. Hence, x V (ju n 4) = x v (/~' N 4); so/~ = # ' , b y 17.4. This completes t h e proof.
N e x t we a t t e m p t to define a filtration on L, b u t we work ' d o w n f r o m 1' r a t h e r t h a n ' u p f r o m zero'. Suppose t h e n t h a t w EL is such t h a t there exists in L a chain:
17.6 W = W k ' < ( W k _ l ~ ... " ~ W l - ~ 1.
The least such/c will be d e n o t e d b y dL(w) w i t h t h e subscript often omitted: clearly if w EL*
w h e n d(w) is defined a n d d(w) = 1. W e e x t e n d t h e definition of d to all L b y setting d(1) = 0, with d(w) = ~ if d(w) is n o t finite. T h u s we h a v e a function d: L ~ A = ( 0 ) U N U ( ~ } ; t h e corresponding function
da.L:
a . L ~ A , for a n y a EL, will be a b b r e v i a t e d to d~. A n easy con- sequence of the definitions is:17.7 LEMMA. 1 / a eL* and d~(w) < 0% then d(w) < oo and da(w ) >~d(w)-1.
To prove a converse (and for other purposes) we n o w impose an e x t r a condition on L.
A~. I / d ( w ) < co then w.L. satis/ies )'1, and (w.L)* ~ 0 i / w =4=0.
17.8. LEMMA. Suppose thatL satis/ies A ' 1. IT/a EL* and d(w) < co then
da(w )
=d(w) - 1 < ~ ,(w<a).
Proo]. Suppose d ( w ) = k < 0% so t h a t there is a shortest chain as in 17.1 above. W e t h u s h a v e a chain (in a.L):
(i) w = a n w k <~anwk_l ~ . . . < ~ a n w l <a.
I f a = wl, there is n o t h i n g to prove. I f a =t = wl, t h e n b y A 1 for L, a N w 1 ~ a, a n d a N w 1 E (wrL)*.
2 3 2 H . B . GRIFFITHS
S u p p o s e i n d u c t i v e l y t h a t a N w~ ~,a N wj_l M ... -< a N wl-< a a n d a N wj E (wrL)*, or a N wj_ 1 = w r N o w d(wj) < ~ since 17.1 gives a f i n i t e chain, i n L f r o m wj t o 1, so b y A'I, w r L satisfies A 1 a n d h a s co-atoms. Thus, since a N wj a n d wj+ 1 b o t h lie in (wrL)*, t h e n e i t h e r a N wj = wj+l, or
a N wj+~ = (a N wj) N wj.~ 6 (a N w~.L)* N (w~+~.L)*
w h e n c e a N w j + ~ , a N wj a n d a N wj+ 1E (wj+rL)*. This justifies t h e i n d u c t i v e s u p p o s i t i o n , a n d p r o v e s t h a t t h e r e is a c h a i n in a.L:
w = wk-< ....~wj+l.<a N wj_v'<a N w j _ 2 ~ . . . ' < a N w l ~ a
w i t h k - 1 t e r m s ; if t h e e q u a l i t y a N wj_ 1 = w] n e v e r o c c u r r e d t h e n in p a r t i c u l a r we w o u l d h a v e b o t h a N wk_ 1 a n d w k in (W~_rL)* w i t h wk<~a N wk_ 1 so w k - a N wk_ r T h u s we c o u l d s t a r t t h e c h a i n w i t h wk = a N wk_l ~, ... which is a g a i n of l e n g t h k - 1, a n d t h e p r o o f is com- plete.
17.9 COROLLARY. I / d ( a ) < ~ , then d ~ ( w ) = d ( w ) - d ( a ) , (w<~a).
(For, a E (b.L)* for some b w i t h d(b) = d(a) - 1.)
17.10 COROLLARY. I / U>~V in L then d(u) <~d(v), provided d(u) < ~ . (For, v e u . L a n d 0 ~du(v) =d(z) - d ( u ) ) .
17.11. COROLLARY. 1 / a E L * and d ( O ) = m < ~ , then da(O ) = m - 1 .
17.12 PI~OPOSITION. L e t L s denote the subset o / L , consisting o / a l l x o / / i n i t e d e p t h . I / d(O) < ~ then L F is a sub-izoposet o / L .
Proo/. C l e a r l y 0 a n d 1 lie in Lp, so i t r e m a i n s to p r o v e t h a t if u, v6L~, t h e n d(u N v) < ~ . T h i s is clear, if d(0) = 1 or if
d(u) +d(v) = 1.
S u p p o s e n o w t h a t t h e p r o p o s i t i o n holds for all i z o p o s e t s M w i t h dM(O)<alL(0) a n d for all u', v' in L w i t h d(u')§ I f u, v in L s a t i s f y d ( u ) § we m a y s u p p o s e n > l ( r e m a r k e d above); if d ( u ) = d(v)= l , t h e n d(u N v ) = 2 b y c o n d i t i o n A 1. S u p p o s e t h e n t h a t d(v) >~2, so t h e r e e x i s t s w EL such t h a t v ~ , w ~=1, a n d 0 < d ( w ) < d ( v ) - i < ~ . T h e r e f o r e d ( u ) + d ( w ) < n so d(u N w ) < ~ . I n w . L we h a v e , b y C o r o l l a r y 17.9, t h a t dw(O)=dL(O)- d(w) <dL(0) so b y t h e i n d u c t i v e h y p o t h e s i s a p p l i e d t o w.L,
dw((u N w) n v) < oo
whence d(u n v) =dw(u N v) § < oo b y 17.9 again. This c o m p l e t e s t h e proof, b y i n d u c t i o n . N e x t we s t r e n g t h e n our c o n d i t i o n s f u r t h e r b y a s s u m i n g t h a t L satisfies t h e following
T H E H O M O L O G Y G R O U P S OF SOME O R D E R E D SYSTEMS 233 c o n d i t i o n , w h i c h h o l d s in t h e d u a l F ~ of a n y f i n i t e g e o m e t r i c l a t t i c e F ( s i n c e A~ in P ~ is t h e c o n d i t i o n of s e m i - m o d u l a r i t y in F):
A 3. I[ d(w)< c~, then w.L satis/ies A 1 and A 2 (with (w.L)* 4 0 i / w 4 0 ) . (Note: w.L h a s a t o m s if L has, a n d (w.L). = w . L . . )
17.13 LEMM~.. I] d(O) = k + l < ~ , then d(a) = k / o r each atom a E L . .
Proo]. T h e r e is a c h a i n O-<Wk+V<Wk'<...'<Wl~I in L, where wk+l=b m u s t be a n a t o m . C e r t a i n t l y t h e n , d(b)<~k; a n d we c a n n o t h a v e d(a)<k for a n y a t o m a, o t h e r w i s e (since 0-<a) d ( 0 ) < ~ d ( a ) + I < k + 1. H e n c e d(b)= k. N o w l e t a b e a n a t o m , a 4 b. T h e n t h e r e e x i s t s a g r e a t e s t ] such t h a t a~wj+l, a Ewj (wo = 1 ) .
T h u s wj+2 E
(wT+i.L)*
in wj.L, a n d a E (wj.L)., so b y L a a n d L e m m a 17.1 for wrL, we h a v e a V wj+2~w r Also b y L e m m a 17.3 for wrL, wj+ 2 = (a V ws+2) n wj+ 1E (a V wj+2.L)* b y A1.N o w s u p p o s e i n d u c t i v e l y t h a t a v wj+~(a V wj+~_l ~... ~ a V wj+~-<wj a n d wj+p-<a V wj+~.
T h e n b y L e m m a s 17.1 a n d 17.2 w i t h x, ~, 2 t a k e n t o b e a, wj+p+~, wj+~ in a V w j . , . L we g e t a Vwj+~+l~a Vwj+~ a n d wj+~+l = ( a Vwj+~+l) N wj+~-<a Vwj+,+l, j u s t i f y i n g t h e i n d u c t i v e s u p p o s i t i o n , 1 ~ p ~ ]c § 1 - ] . H e n c e we h a v e a c h a i n
a'<a V wk+l ~ ...-~a V w~+u'~ w~'< ... "< 1, so d(a) <~/c. W e o b s e r v e d t h a t d(a)~]c, so d(a) =It as r e q u i r e d .
17.14 LEMM)~. I] d(w)< oo, and a ~ L . , then a Vw is de/ined: i/ a~w, then a V w = w ; i/
a Cw, then w ~ (a V w.L)*, and d(a V w) =d(w) - 1.
Proo/. I f d(w) < 2 , t h e n t h e s t a t e m e n t h o l d s b y A2 a n d C o r o l l a r y 17.4.
S u p p o s e t h e n t h a t d(w) = k so t h a t t h e r e is a c h a i n w ~ w ' ~ w " - < . . . ~ 1. C l e a r l y d(w') <~
k - l , a n d if d ( w ' ) < k - 1 t h e n d(w)=d(w')+1 < k . T h u s d ( w ' ) = k - 1 .
T h e r e f o r e b y a n i n d u c t i v e h y p o t h e s i s , v = a V w' is defined, d(v) = k - 2, a n d b o t h a a n d w' lie in v.L = M. M o r e o v e r ( b y t h e h y p o t h e s i s ) w' ~ M*. Thus, a V w is d e f i n e d since M satis- fies A~ (for L satisfies An). I f a r w, t h e n b y L e m m a 17.1, a V w ~ M*, a n d b y C o r o l l a r y 17.3, w ~ (a V w.L)*. Also b y C o r o l l a r y 17.4, d(a V w) = dv (a V w) + d(v) = 1 § (k - 2) = d(w) - 1 a n d t h e l e m m a follows b y i n d u c t i o n .
17.15. L E M ~ A . I n the set L~ o/elements o / L o/finite depth, we have (w.L~)*'~L~,4~, pro- vided w ~Lr and w.Lr ~ (w}.
Proo/. Since w.L~ =~(w}, t h e r e e x i s t s v ~wL~ w i t h v < w s u c h t h a t a <~w. I f v is n o t a co- a t o m of w.L~, t h e r e e x i s t s u e w . L r w i t h v < u < w . N o w b y 17.9, d~(u) exists, so 0 < d w ( u ) = d ( w ) - d ( u ) , w h e n c e d(v)>d(u)>d(w). T h e r e f o r e we r e a c h a c o - a t o m y of w.L~ f r o m v a f t e r a t m o s t dw(v) - 1 steps, a n d d(y) < co so (w.Ly)* f) L F ~=~.
1 6 - - 722902 A c t a mathematica 129. I m p r i m 6 le 5 O c t o b r e 1972
2 3 4 H . B . G R I F F I T H S
17.16 C O R O L L A R u OF e a O O F . I f u < W E L F , then d(w)<d(u); if d(u) + 1 (because dw(u) = 1).
W e can n o w e x t e n d 17.12 to:
u-< w then d(w) =
17.17 THEOREM. I f d(O) < ~ in L, and L satisfies As, then L F is a sub-izoposet o / L which has the same atoms as L and which also satisfies A s.
Proof. W e k n o w f r o m 17.12 t h a t LF is a sub-izoposet, while b y 17.13, L . = (Ls).. Also, b y 17.15, if 0 < w ELs, each set (w.LF)* , t a k e n relative to LF, is n o n - e m p t y . I f ~ 4 # in (w.LF)*
t h e n b y A~ for w.L, ~ N # E ( ~ L ) * N (#L)*; therefore ), N/~-<X, so ~ N # E L F b y 17.16, whence )~ N/~ E (X.LF)*. This verifies A'I for Ls. To show t h a t w.Ls satisfies A2 we proceed similarly, using 17.14. Therefore L F satisfies A s as required.
N o w recall conditions Le - L 5 used in previous sections. L e t us denote b y Ls89 t h e con- dition L 4 with t h e clause a b o u t cardinals deleted.
17.18 T H ]~ o 1~ n M. I / L is an izoposet satisfying A s such that every element is o / f i n i t e depth, then L is a lattice and satisfies L 1 -L3~.
Proof. B y 17.16, the function r is a filtration on L, where r a n d r is minimal (i.e. L2 holds). B y 17.14, condition L a holds; so L 1 holds a n d L is a lattice as we ob- served prior to 10.2. Since 17.5 holds in each set w.L, b y A 3 a n d 17.15, t h e n La~ follows.
Thiscompletes t h e proof.
A consequence of this t h e o r e m is t h a t we can forget a b o u t those mysterious elements of L which h a v e infinite depth, when c o m p u t i n g t h e h o m o l o g y of ~FgX in 4.1. For, we saw in section l l , t h a t starting with the function g: ( I ) X ~ L we could pass to m: ~ 9 A ~ L in 11.2; a n d we can change L t o t h e image of m, which is generated b y atomic sums a n d hence lies in L~ if L satisfies A s. Therefore, we can change f i r m L to Lx, which then satisfies Condi- tions L 1 - L 5.
As a sort of converse of 17.18, we prove the n e x t result which leads to a curious con- clusion a b o u t geometric lattices (see 13.6).
17.19. THwOR]~M. Let L be a submodular lattice with 0 and 1, satisfying L 2 and L s. T h e n the dual, L*, o / L satisfies A s and all elements o/L* are o / f i n i t e depth. Further, all m a x i m a l
chains in L* between fixed end-?oints are o/constant length.
Proof. The strict m o n o t o n i c i t y of r excludes infinite chains in L, so all elements x EL h a v e finite d e p t h d(x), and x has d e p t h r - d ( x ) in L*. B y L2, t h e length of a n y m a x i m a l chain in L (or L*) between x a n d y is It(x) - r I" Since there are no infinite chains, co-
THE HOMOLOGY GROUPS OF SOME ORDERED SYSTEMS 235 Ox/ord Con]erenee on Combinatorial Mathematics and its applications, Academic Press, 1971.
Received July 17, 1970
Received i.r~ revieed ]orm February 13, 1972