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The Construction of Some Free m-Lattices on Posets
Annals of Discrete Mathematics 23 (1984) 103-118 0 Elsevier Science Publishers B.V.(North-Holland)
103
THE CONSTRUCTION OF SOMF FREE m-LATTICES ON POSETS George G G T Z E R and David KELLY Department of Mathematics University of Manitoba W i nn i peg, Man i t o ba CANADA Dedicated to E . Corominas
11 est bien connu que les treillis peuvent letre considgrgs, soit comme d e s ensembles ordonnes, soit comme des algsbres universelles. L'approche alg6brique
garantit l'existence d'un treillis libre sur tout ensemble ordonn6 ; c'est-2-dire du treillis engendr6 "le plus librement possible" B partir de l'ensemble ordonn6. Beaucoup des ensembles ordonn6s qui apparaissent naturellement dans les differrntes branches des math6matiques ant une structure de treillis. En fait ce sont habituellement des treillis complets. Cependant, quelques techniques alg6briques ne s'appliquent plus directement aux treillis complets. En particulier,
il n'y a pas de treillis complet libre sur un ensemble B trois 6lQments. Le concept
B la th6orie de l'ordre, tandis que la notion de treillis
de complgtude appartient
libre sur un ensemble ordonne provient de l'approche alggbrique. L'inexisterice de treillis complets libres montre qu'on doit-gtre prudent en essayant de combiner ces deux concepts. Nous rgsolvons cette difficult6 en affaiblissant le concept de compl6tude. un cardinal r6gulier infini. Un treillis m-complet ou m-treillis
Soit m
est un ensemble ordonnQ dans lequel toute partie non vide ayant mains de m.El6ments possede une borne supgrieure et une borne infgrieure. Chaque concept de la thgorie des treillis s'dtend en un concept correspondant dans la th6orie des m-treillis. On omet
m
quand
m
=
K,
car c'est de la th6orie des treillis. I1 est clair
qu'un treillis complet est m-complet pour tout m que l a cardinalit6 d ' u n ensemble ordonnh existe dans
P
. De
plus, si m
P , alors toute borne supgrieure qui
est la borne sup6rieure d'une partie non vide de
dinalit6 est inf6rieure 2
m
est plus grand
P dont la car-
; en bref toute borne sup6rieure est une m-borne
sup6rieure. Cependant, i l existe des m-treillis qui sont complets et dans lesquels toute borne sup6rieure n'est pas une m-borne supgrieure. Soulignons que n o u s prenons uniquement les bornes sup6rieures et inf6rieures d e parties non vides. Bien que les m-treillis aient des opgrations infinitaires, le nombre d e ces opgrations e s t born6 de sorte que les concepts et les mdthodes de l'alggbre universelle s'appliquent. Le point important est qu'il existe un ensemble (non une classe) qui indexe simultan6ment toutes les op6rations
G. Gratrer, D. Kelly
104
bornes superieures et inf6rieures dans chaque m-treillis. Une cons6quence particuliGre est l'existence, p o u r tout ensemble ordonne
P
de
Fm(P),
le
P. F,(P) est l e m-treillis qui contient P comme S O U S m-treillis libre sur ensemble ordonne pour lequel les m-bornes sup6rieures et m-bornes infcrieures sont formees "aussi librement clue possible". (En particulier, ). Plus
P
dr6 par
application isotone d e longement B
F (P)
e s t m-engen-
Fm(P)
F ( P ) est l'unique m-treillis pour lequel toute m dans n'irnporte quel m-treillis L a un unique pro-
prgcisgment, P
qui est un
m-homomorphisme.
Un ensemble ordonne est m r n s'il ne contient aucun d e s trois ensembles I 2 3 1 5 + + ; I , "v + 2. , 'L + 2. . I. RIVAL et R. WILLE ont dGcrit 'L le treillis libre sur tout ensemble ordonne mince. Nous avons Etendu ce resultat
ordonnes suivants
~
F (P) p o u r tout ensemble ordonn6 P . Comme d a m m l'article de I. RIVAL et R. WILLE l'argument cl6 dans la description est la d e s -
aux m-treillis en d6crivant
cription du m-treillis libre sur l'ensemble o r d o n n e
< a2 < a l ,
definie par .a denombrable, m
=
He)
.
F
m
bo
< b2 < b , ,
.a
H = {ao,a, ,a2,bo,bl,b2}
< bl
bo
< at .
Pour
m
non
(H) est un treillis complet (ce qui n'est p l u s vrai lorsque
Dans cet article de synthsse, n o u s decrivons F (H) m
et
Fm(H)
.
C o m e le m-treillis
est assez compliqu6, il ne peut pas stre appr6hend6 immgdiatement. Nous
donnons beaucoup de diagrammes pour expliquer l e s formules donnant les bornes
sup6rieures et inferieures dans
F,(H)
.
Une motivation intuitive est aussi donnee
p o u r beaucoup de ces formules. La preuve que le rn-treillis dgcrit est
Fm(H)
est
publi6e ailleurs.
Abstract
Let an
rn be an i n f i n i t e regular c a r d i n a l . We c a l l a Loset m-ccnplete l a t t i c e (or m l a t t i c e ) i f every nortenipty
m elements has both a j o i n and a subset with fewer thsn nieet. This concebt was introduceu by P. Crawley and h.A. Lean. For m - l a t t i c e s , enough of the a l g e b r a i c f l a v o r of the usual ( f i n i t a r y ) l a t t i c e case has been retained so t h a t Lie there are f r ee m - l a t t i c e s generated by any poset. s h a l l d e s c r i b e the f r e e m l a t t i c e on a p a r t i c u l a r 6-element poset 11, and i n d i c a t e w h y t h e d e s c r i p t i o n is c o r r e c t . l'he poset ii is a n a t u r a l example because i t s f r e e m-latt i c e can be used t o describe t h e f w e m - l a t t i c e on many other posets. IHlHOCUCTIOII
I t is well known t h a t l a t t i c e s can be considered e i t h e r a s ordered s e t s or a s universal algebras. The a l g e b r a i c approach guarantees the e x i s t e n c e of t h e f r e e l a t t i c e on any p s e t ; t h a t i s , the l a t t i c e t h a t is kenerated "as f r e e l y a s possibletr s t a r t i n g from t h e poset. ( I f no ordering is specif'ied on t h e gerlera t i n g p o s e t , 2n antichain is undtrstood. )
105
Some free m-lattices on posets
Eiany of t h e ordered s e t s t h a t n a t u r a l l y occur i n various f i e l d s of n:stkienl&t i c s have a l a t t i c e s t r u c t u r e . I n f a c t , they a r e usually com.~lete l a t t i c e s . llowever, sorile e l g ~ b r a i ctechniques no l o n t e r apply d i r e c t l y t o complete l a t t i c e s . I n p a r t i c u l a r , t h e r e is no f r e e complete l a t t i c e on t h r e e elen,ents. If such a complete l a t t i c e d i d e x i s t , then every coicblete l a t t i c e L w i t h t h r e e g e n e r a t o r s would be a complete homoniorpliic image of i t ; i n p a r t i c u l a r , t h e r e would be ari L. llowever, E'iLure 1 shows a upper bound f o r t h e c a r d i n a l i t y of any such 3-generated cornblete l a t t i c e of any given c a r d i n a l i t y . (In t h i s f i t u r e , the t h r e e conlplete generators a r e denoted by squares. ) O f c o u r s e , any diagrem of an i n f i n i t e poset r e q u i r e s i r i t e r p r e t a t i o n . Ari extreme p o s i t i o n is t h a t any such diaLran, is u s e l e s s . h'e a r e not of' t h i s opinion, b u t we do advise extreine c a r e , e s p e c i a l l y i n a proof, when r e a d i n t information fron, a ciisgran; of a i i n f i n i t e poset. b'or example, it is not obvious when an element is j o i n i r r e d u c i b l e . (F'or a f i n i t e l a t t i c e , j o i n i r r e d u c i b l e elements a r e those w i t h a unique lower cover.)
The left-hand chain C i n Figure 1 can be ariy well-orderec chain t h s t bias a l a r b e s t element, denoted by t . The r i ~ h t - h a n d choin is isomorphic t o C , arid t h e r e m i n d e r of t h e poset is a l i n e a r sunl of f i n i t e l s t t i c e s over C . For eacki i t i n C , t h e poset of Figure 1 tias an i-th l e v e l c o n s i s t i n g of' 6 elenients. i < t can only be generatea a f t e r , ari Observe t h a t an elenerit on any l e v e l element on every lower l e v e l has beeti t e n e r a t e d . The concebt of completeness belongs t o order theory. while f r e e l a t t i c e s or1
FIGURE 1 . An a r b i t r a r i l y l a r g e c o m p l e t e l a t t i c e
POSetS come from t h e a l g e b r a i c approsch.
The above example shows t h a t
One
must
G. Gratzer, D. Kelly
106
be c a r e f u l when attempting t o combine these two concepts. d i f f i c u l t y by weakening t h e concept o f completeness.
We s h a l l resolve this
We s h a l l always use t h e symbol m t o denote an i n f i n i t e r e t u l a r c a r d i n a l . An m-complete l a t t i c e , o r i i - l a t t i c e , is a poset i n which every nonempty s u b s e t with fewer than m elements has both a j o i n and a meet. We s h a l l pre fix each l a t t i c e concept with m t o denote t h e Corresponding concept f o r n - l a t t i c e s . Ue m i t m Some examples a r e in-join, ll.-hornornorphism and ni - s u b l a t t i c e . when m :to because t h i s i s t h e usual l a t t i c e case. Obviously, any complete l a t t i c e is an .i-lattice for any ( n . Moreover, P , then any join t h a t e x i s t s i n P i s a j o i n of a nonempty subset of P whose c a r d i n a l i t y is less than .it; i n s h o r t , However, t h e r e a r e it-lattices t h a t a r e complete in any jo i n is an m-join. which not every j o i n is an m j o i n . I f the left-hand chain C i n Figure 1 is t h e ( o r d i n a l ) successor of t h e i n i t i a l o r d i n al corresponding t o m, then t h i s f ig u r e provides such an example L. I n this cas e , note t h a t t h e top element t of C cannot be w r i t t en a s V S f o r a nonempty subset S of L with t L S and IS1 < m. In o t h er words, t is an m-join i r r e d u c i b l e eleuient of L. Consequently, t h e t h r e e squares i n Figure 1 do not m-generate L. if
n exceeds t h e c a r d i n a l i t y of a poset
We emphasize t h a t we only t ak e j o i n s and meets of nonempty s e t s . (In a complete l a t t i c e L , t h e j o i n of t h e empty s e t equals t h e meet of L.) Although m - la tti c e s have i n f i n i t a r y o p er at i o n s , t h er e is a bound on t h e number of such operations so t h a t t h e concepts and methods of universal algebra apply. The important point here is t h a t t h er e is a (not a c l a s s ) t h a t can simultaneously A particular index a l l t h e j o i n and meet o p er at i o n s of every m - l a t t i c e . consequence is t h e ex i s t en ce, f o r any poset P, of F,(P), t h e free m - l a t t i c e
set
on
P.
Fm(P)
is t h e
m-lattice
t h a t co n t ai ns
P
a s a subposet f o r which
m-joins and m-meets a r e formed "as f r e e l y a s possible". (In particular, F,(P) is al-generated by P . ) More p r eci s el y , F A P ) is t h e unique m - l a t t i c e f o r which every isotone map from P t o any m-la ttic e L has a unique extension t o an m-homomorphism from F AP ) t o L. W e s h a l l use n_ , a boldface number, t o denote an n-element chain. A l i g h t f a c e number i n d i c a t e s an a n t i chain. A s u s u a l , P + Q denotes t h e d i s j o i n t union of t h e posets P and Q. We s h a l l c a l l a poset slender i f it does not contain any of t h e following th r e e p o s e t s a s a subposet: (i)
(ii) (iii)
3=1+1+1 2, + 1+
2 5
.
I. Rival and R. Wille [ I 6 1 described t h e f r e e l a t t i c e on any f i n i t e slender poset. I n a sequence o f papers "91, [ l o ] , [ill, [121), we have generalized t h i s r e s u l t by describing F-(P) f o r any slender poset. Our techniques a l s o work when m is countable, and so provide another proof of t h e Rival-Wille r e s u l t . However, since t h e rn A case d i f f e r s s i g n i f i c a n t l y from a l l o t h e r s , we
henceforth assume t h a t
ni
is uncountable.
S i m i l a r l y a s i n [16], t h e key ingredient i n describing t h e free m - l a t t i c e on any slender poset is a d es cr i p t i o n of t h e f r e e m -la ttic e on t h e poset H of Figure 2. The present paper is devoted t o describing Fm(H). We s h a l l se e t h a t
is a complete l a t t i c e (which would not be t r u e i f we allowed m F,(H) In g e n e r a l , we s h a l l only i n d i c a t e proofs. W e not only de sc ribe F,(R)
No). as a
Some free m-lattices on posets
107
p o s e t , but we L i v e t h e formulas !'or ( a r b i t r a r y ncnenpty) j o i n s and meets.
FIGURt 2. The Loset
I n order t o d e s c r i b e
11
Fm(fl), we mst f i r s t introduce sonle o t h e r
n-lat
t i c e s . I n Section 1 , we describe C ( m ) , t h e f r e e m - l a t t i c e on 2 + 2 , w i t h a new 0 and 1. We a l s o r e q u i r e t h e m - l z t t i c e s A and b oescribed i n Section 2. I n Section j , we d e s c r i b e D(m 1, which is Fm(H) with a new C, and 1 . Our paper [91 contains ttie proof's of a l l s t a t e n e n t s nade i r i t h e f'irst three sections. I n Section 4 , we d i s c u s s the p-oofs t h a t t h e m-lattices C ( m ) and C( m ) , a s d e s c r i b e d , a r e what we claimed. One approach w i l l be t o apply t h e S t r u c t u r e Theorem f o r f r e e m - l a t t i c e s on posets given by P. Crawley and Fi.A. Dean [21. Complete d e t a i l s f o r Section 4 can be found i n L91 x i a [ l o ] . Let P be a slender poset. I f F is a l s o l i n e a r l y indecoriiposable, then we F is countable, and t h a t Fm(P) can be embedded, as s n show i n [ill t h a t m-sublattice,
into
does not contain
F,,,(H).
Consequently,
a poset
F is slender i f f
Fm(P)
Fm(3) a s an m-sublattice.
W e conclude t h i s introduction by b r i e f l y mentioninb some f u r t h e r t o p i c s about m-lattices. Free m - l a t t i c e s , defined a s f r e e m - l a t t i c e s over a n t i c h a i n s , can a l s o be considerea a s f r e e ?.-products of one-element m-lattices. For f r e e products, we r e f e r t o G r a t z e r , Lakser and P l a t t [ I 4 1 arid Gratzer and Lakser [ l 3 ] . I n our paper 171, we Generalized sonJe o f t h e r e s u l t s o f ' t h e s e two papers, and a l s o studied what happens when m is v a r i e d . For example, we proved t h e S t u c t u r e Theorem f o r f r e e m-products, which is s i m i l a r t o t h e S t r u c t u r e Theorem f o r f r e e m - l a t t i c e s of Crawley and Dean [21. Another r e s u l t o f [ T I is: I f a covers b i n some f r e e m -product, then a continues t o cover b when m is increased. (Covers a r e of interest becsuse A. Cay [31 has shown t h a t every proper closed i n t e r v a l of a f i n i t e l y generated f r e e l a t t i c e contains a covering p a i r . ) We a l s o determined which f r e e m-products a r e complete i n [TI. I n p a r t i c u l a r , t h e f r e e " - l a t t i c e on t h r e e g e n e r a t o r s i s not complete ( s e e Whitnian [1&1 f o r t h e m = case). When r e p r e s e n t i n g an element of an m l a t t i c e i n terms of an m -generatink set (using m-Joins and m-meets), t h e r e is no general way t o s p e c i f y a unique r e p r e s e n t a t i o n , even i n a f r e e m - l a t t i c e . For example, t h e r e a r e many exaniples in
Fm(3)
of an
m-join
reducible element t h a t cannot be expressed a s an
irredundant m-join. In 171, we generalized t h e Normal Form Theorern t h a t 6 . Jdnsson [151 proved f o r t h e f r e e m - l a t t i c e on a poset. We used t h e t h e norm61 r e p r e s e n t a t i o n o f [7] t o g e n e r a l i z e a r e s u l t about f r e e products i n 1131.
A n a t u r a l question about an ordered s e t is: M a t is t h e l a r g e s t chain t h a t
it c o n t a i n s ?
We say t h a t a poset P c o n t a i n s no chain of c a r d i n a l i t y n . r e g u l a r . M.E. Adams and 0 . K e l l y [ l ]
s a t i s f i e s the n-chain condition i f it Henceforth, l e t n 6 e uncountable and showed t h a t , i f t h e poset P s a t i s f i e s
G. Gratzer, D. Kelly
108
the n-chain c o n d i t i o n , then so does the f r e e l a t t i c e on P. However, f o r m, we contructed, i n our paper [61 with A . llajnal (for any uncountable n > m ) , a poset P with no uncountable chains such t h a t t h e f r e e m - l a t t l c e on P contains a chain of c a r d i n a l i t y n . I n Adam and k e l l y [ l ] , it was a l s o show t h a t t h e f r e e product construction preserves t h e n -chain condition. In our paper [61 with A. Hajnal, we a l s o showed t h a t a f r e e m - l a t t i c e cannot contain a chain with more than m elements. ( I n t h e previous sentence, we assumed t h e Generalized Continuum Hypothesis t o simplify t h e formulation.) If m gl and n is t h e successor c a r d i n a l t o t h e continuum, then we do n'ot know whether the f r e e
1.
m-product construction preserves t h e n-chain c o n a i t i o n .
THE COMPLETE LATTICE
C( m)
Recall t h a t m always denotes an uncountable r e g u l a r c a r d i n a l . The complete l a t t i c e C ( m ) of Figure 3 is t h e f r e e m-lattice on 2, + 2 with a new 0
and
1.
F(2 +
The diagram is a l s o v a l i d f o r
2)
( s e e Rolf [Yl or Chapter VI
of Gratzer [51). W e have denoted t h e 16-element l a t t i c e i n t h e middle of Figure 3 by C( A o ) . (The way t h i s f i n i t e l a t t i c e was used by I. Rival and R. Wille [I61 t o c o n s t r u c t F(H) is s i m i l a r t o t h e way we s h a l l use C ( m ) F A H ) .I I n t h i s s e c t i o n , we only d i s c u s s t h e complete l a t t i c e c( m). We leave u n t i l Section 4 t h e question whether C ( m ) is claimed. Observe t h a t a,, a;, b,, and b h generate C ( K O ) . t h e notation f, a A b and f ' a ' v bh. n m m m
t o construct s t r u c t u r e of what we have W e a l s o use
For every successor o r d i n a l j < in, t h e r e is a lower j-th l e v e l of 6 elements L. l a . , b . , c ., d ., e j , f j ) , and f o r every l i m i t o r d i n a l i < M J J J J J (including i 01, t h e r e is a lower i-th l e v e l of 7 elements Li = { a i , b i , c i , d i , e i , f i , g i ] . These elements a r e ordered a s shown i n Figure 3. There i s a l s o an upper i-th level
f o r each
Ui
i
m, defined d u a l l y and denoted by t h e safile
<
l e t t e r s with primes. For convenience, we a l s o l a b e l 6 elements of ah, p bO, p' bb, y go, y ' = gb. Greek l e t t e r s : a I ao, a '
I n a more formal approach, one d e f i n e s
c( so) uu ( LJ. I
j
C( m
t o be
LI ( uJ.
I j
C(
m)
with
where each f i n i t e poset t h a t we have l i s t e d i s assumed t o be a s u b p s e t of C( T ) . Let i and j be o r d i n a l s l e s s than m The a d d i t i o n a l comparability u < v holds i n C ( m ) e x a c t l y when one of t h e following conditions is s a t i s f ied :
.
u = a i , v = a . and i < j ; J' u = bi, v = b . , and i < j ; J u E Li [ a i , b i l , v E L . , and i J u = ai and a, 5 v i n C ( % o ) ;
-
u
= b.1 and b,
-
5
v
in
C( E'o)
<
j ;
;
E Li { a i , bil and v E C ( g o ) or v E :U J' a previous condition holds w i t h u and v interchanged, L and U interchanged, primed elements, and r e v e r s a l of t h e i n e q u a l i t y i n
u
C(
so);
Some free m-lattices on posets
109
i limit
a:
Ago= y
FIGURE 3.
The complete lattice
C( m 1
G. Gratzer, D. Kelly
110
u
and
*
v
b!; J u = bi and v E U . but v :a!. J J' Clearly, y go is t h e least element of C ( m ) , and y' = pi, is t h e is a complete l a t t i c e . To sketch t h e proof o f t h i s g r e a t e s t . In f a c t , C ( " ) s t a t e m e n t , l e t u s e x p l a i n how t o c a l c u l a t e (nonempty) j o i n s . From Figure 3, one sees a n a t u r a l way to break C ( m ) up i n t o 7 p a r t s : C ( AO), the bottom l e f t chain [ a i I i 5 m I , t h e bottom middle p a r t from go t o k , t h e bottom a.
1
v
U.
E
J'
but
r i g h t chain bi j i 5 m I , and t h e corresponding t h r e e t o p p a r t s . C( has e x a c t l y one p o i n t i n common with each of t h e o t h e r six p a r t s . Observe t h a t C( m ) . each p a r t is a complete l a t t i c e , i n f a c t , a complete s u b l a t t i c e o f Since each i n f i n i t e p a r t is a l i n e a r sum over a complete c h a i n , it should be c l e a r how t o c a l c u l a t e t h e j o i n i n each p a r t . I t only remains t o determine t h e W e leave the j o i n o f p a i r s of incomparable elements frorn d i f f e r e n t p a r t s . d e r i v a t i o n of t h e s e formulas t o t h e r e a d e r .
8'.
Let u s i n d i c a t e why C( m ) We f i r s t g e n e r a t e elements of
-
[y, y ' ) is m-generated by a , a ' , C ( &o) i n t h e following o r d e r :
since 0 i d e a f o r t h e rest of t h e argument complete l a t t i c e of Figure 1. I f i < m , then it is easy t o generate
We can now generate
a l i m i t ordinal then 2.
a. J
and
C( X )
j
Li
m,
<
we have its four g e n e r a t o r s ,
.
p, and
We kave t h e
i n t h e i n t r o d u c t i o n when we discussed t h e Li has a l r e a d y been m-generated f o r some Li+l using f i n i t e j o i n s and meets. I f , for
has already been =-generated
b . can then be m-generated s i n c e J
a. =
( ai
J
f o r each
I i
<
j)
.
i
<
j,
THE LATTICE A
I n t h i s section, we d e f i n e t h e second b u i l d i n g block of D( m ) , t h e l a t t i c e of F i g u r e 4. This l a t t i c e was discovered by I . Rival and R . Wille L161. Let J be t h e set of dyadic r a t i o n a l s r t h a t s a t i s f y 0 5 r s 1. Every nonzero r E J has a unique r e p r e s e n t a t i o n , t h e normal form, r = a * 2-", where a is an odd
A
integer;
n
is t h e
order of
r; i n notation, n
= 0. Observe t h a t o r d ( r ) = o r d ( 1 maximum of o r d ( r ) and o r d ( s ) . We d e f i n e
A
a s a subposet of
- r). J2
Although
< r , s>
i n Figure 4.
E
A , then
v
o r d < r , s> t o b e t h e
-ord s - r = 2 I .
not
with t h e exception of J 2 , For example,
a s u b l a t t i c e of
<1/4, 1/2> = <1/2, I > A
can be redefined i n two ways:
A = I I r
<
s and
s
[ < r , s> I r
<
s
and
s
A
W e a l s o define
o r d ( r ) ,i ord(s)
is a l a t t i c e , it is
A
Ey convention, o r d ( 0 )
with t h e product o r d e r :
A = { < r , s > 1 r < sand Note t h a t i f
ord(r).
-
r = 2-",
n >. o r d ( r ) l
r = 2-",
n
2
ord(s)}
.
<0, l > .
111
Some free m-lattices on posets
We s h a l l use
and
nl
t o denote t h e f i r s t and second p r o j e c t i o n s of
n2
J2
J . For t E J , l e t us c a l l t h e set of a E A t h a t s a t i s f y n , ( a ) t x = t l i n e i n A , and d e f i n e t h e y z t l i n e similarly. The a l t e r n a t e d e f i n i t i o n s of A y i e l d e x p r e s s i o n s f o r t h e x = r and y s l i n e s i n A. In > is t h e l a r g e s t element on t h e x = r l i n e , and p a r t i c u l a r , < r , r + 2- Ord(') is t h e s m a l l e s t element on t h e y = s l i n e .
onto
the
Each a formulas a r e :
For example, 1>1
denoted by
has a r i g h t lower cover
a*
and a r i g h t upper cover
a*.
The
< ( r + s ) / 2 , s> < r , ( r + s ) / 2 >.
< r , s>* < r , s>, {
A
-
<1/4, 1/2>* <3/8, 1/2> i n Figure 4. Moreover, i f a E A s a t i s f i e s o r d ( a ) = o r d ( n , ( a ) ) , then a has a l e f t lower c o v e r ,
*a, and d e f i n e d by
* < r , s> = < r
-
2-ord(r), s>.
-ord ( s ) > S i m i l a r l y , t h e l e f t upper cover * < r , s> e x i s t s and e q u a l s < r , s + 2 when o r d ( r ) < o r d ( s ) . Observe t h a t <0, 1> has no l e f t (lcwer or upper) c o v e r , b u t t h a t every o t h e r element o f A h a s e x a c t l y one l e f t cover. Moreover, each cover o f an element a E A l i e s on t h e x-line through a or t h e y-line through a .
FIGURE 4.
The l a t t i c e
A
I12
G. Gratzer, D. Kelly
.
2 I n Figure 4 , A seenis t o be a very s p a r s e subset of J For each a = < r , s> E A, l e t us c a l l the closed i n t e r v a l [ r , s ] ( i n e i t h e r t h e r e a l s or dyadics) t h e shadow of a . We say t h a t two closed i n t e r v a l s overlap i f they have an i n t e r v a l of nonzero length i n common, but n e i t h e r i n t e r v a l contains t h e o t h e r . The " r e l a t i v e sparseness'! of A is expressed by t h e following shadow p r i n c i p l e : The shadows of two elements of A cannot overlap. Let u s i n d i c a t e an easy way W e add a v e r t i c a l l i n e ( t h e shadow l i n e ) t h a t passes t o v i s u a l i z e shadows. through t h e i n t e r s e c t i o n of the x = 1/2 and y = 1/2 lines on t h e r i g h t s i d e of A i n Figure 4 ; index t h i s l i n e i n t h e obvious way w i t h 10, 11. I f one places an opaque r i g h t angle a t a E A (forming t h e x - l i n e and t h e y-line through a ) , then t h e i n t e r s e c t i o n of t h e r i g h t angle with t h e shadow l i n e g i v e s t h e shadow of a . W e found t h e shadow p r i n c i p l e t o be a very n a t u r a l t o o l i n our proofs. Let u s apply it t o "explaint1 t h e emptiness of t h e l a r g e s t open square i n Figure 4: i f t h e r e was an element a i n t h i s square, then t h e shadows of a and <0, 1/2> would overlap.
I n Figure 4 , the d i s t a n c e of each element < r , s> E A from t h e shadow l i n e is proportional t o s r , t h e length of its shadow. Consequently, f o r two incomparable elements a , b E A , we can define a t o be t o t h e l e f t of b i f f
-
has a longer shadow than b. Observe t h a t a is t o t h e l e f t of b i f f < n,(b) iff n2(a) > n 2 ( b ) . With t h i s concept, we can v i s u a l i z e t h e 1 f i n i t a r y j o i n i n A. Let a and b be incomparable elements of A , with a t o t h e l e f t of b . The j o i n of a and b is t h e l e a s t element on t h e y - l i n e through a t h a t is g r e a t e r than b. Consequently, a V b an f o r ttie l e a s t n
a
n (a)
such t h a t an > b , where an is inductively defined by: a. T h i s shows t h a t A is a l a t t i c e .
3.
THE COMPLETE LATTICE D(
a , ai+l = ( a i ) *
.
t?t)
W e build up D( m ) from t h r e e d i s j o i n t s u b l a t t i c e s : l a t t i c e A was defined i n Section 2. We d e f i n e B = { < r , s> I < s , r >
E
A , B , and C.
The
A],
a subposet of J2. Clearly, B is a l a t t i c e and its diagram is obtained by r e f l e c t i n g Figure 4 about a v e r t i c a l l i n e . I n o t h e r words, mapping < r , s> E A to E B d e f i n e s an isomorphism of A with B. Consequently, every element has
b
of
B
has l e f t covers
a r i g h t lower cover
b*
or
a
*b
and
*b,
and i f
r i g h t upper cover
b z < 1 , 0 > , then
b
b*.
F i n a l l y , l e t I be t h e r e a l i n t e r v a l [O, 11, and r e c a l l t h a t J denotes t h e s u b s e t of I c o n s i s t i n g of dyadic r a t i o n a l s . For each t E J , we t a k e a and generators a t , a ; , pt, copy Ct of C( m ) , w i t h bounds yt and y t ,
p i , For each t E I two-element chain with
t
E
I.
Since
I
which is riot a dyadic r a t i o n a l , Ct = [y,, y i ] is t h e yt < y.; W e define C a s t h e l i n e a r sum of t h e C t ,
is complete and each
Ct
is complete,
C
is a complete
lattice. To d e f i n e C ( m ) , ( s e e Figures 5 and 6):
we must describe t h e p a r t i a l ordering on
A u b u C
Some free m-lattices on posets
FIGURE 5.
The complete l a t t i c e
FIGURE 6.
Cetails of
fi( m )
113
D(
rn)
G. Gratzer, D. Kelly
114
If
S>
E
< r , s> < r , s> < r , s> < r , s>
< t ,u> < t , u>
A, <
> <
> <
>
< t , u> E B, v E I , and < t , u> i f f s < u; < t , u> i f f r > t ; p i f f s < v holds o r s p i f f r > v holds o r r p i f f t < v holds or t p i f f u > v holds o r u
I t is e a s i l y seen t h a t
Cv,
then:
v v
and
av
and
v
and and
a' 2 V
p
= = = =
E
v
is a poset.
D(m)
_c
pv p{
5
p
hold;
p
hold;
p
hold; hold.
c
2
However, since <0, 1/2> 1 < 1 , 1 / 2 > , v was assumed t o be a dyadic pv, o r p.; Tnus, i f v is not
2.
A u B is not a subposet of J Note t h a t r a t i o n a l wherever we used t h e notation a v , a ; ,
dyadic, then i n each of the l a s t four c a s e s , only t h e f i r s t c l a u s e can apply.
I t is not d i f f i c u l t t o show t h a t C( m ) is a l a t t i c e , and t h a t each of A , B, and C is a s u b l a t t i c e of D( m ) . For < r , s> E A , < t , u> E B , v E I , and p E C v , we give t h e formulas f o r joining p a i r s :
< r , s> v p
(a)
( i ) as v p
is E
(ii) < r , s>,
(b)
C , where t h e j o i n is formed i n
if
r
>
v
r
or
v
w
and
(iii)
t h e l e a s t such t h a t w > v , i f
(iv)
t h e l e a s t such t h a t w t v , i f
p
C, i f
r r
5 5
s
5
v v
<
s and
< s
and
< r , s> v < t , u> i s ( i ) < t , u>, i f s < u; ( i i ) < r , s>, i f t < r ; ( i i i ) the l e a s t on t h e y s l i n e i n A such t h a t w ( i v ) t h e l e a s t < t , w> on t h e x = t l i n e i n B such t h a t w ( v ) as V ps, i f s = t , where t h e j o i n is formed i n Cs. There is an automorphism B, and interchanges a t and
E
@
pt
of
(a;
D( m )
and p i )
v;
i n Cv;
5 a;
t h a t maps f o r any
p .k a; i n Cv; p 5 i n Cv;
> >
t , i f s >t; s, if s >t;
< r , s> E A t o t E J . In o t h e r
words, $ r e f l e c t s Figure 5 a b u t i t s c e n t r a l a x i s . With t h i s observation, t h e formulas f o r < t , u> v p follow from ( a ) . The pairwise meet formulas follow by duality. In order t o show t h a t D( m ) is a complete l a t t i c e , it s u f f i c e s t o f i n d X f o r any nonempty subset X of A. (The formula is s i m i l a r f o r B and we already know t h a t C is complete.) Let X, arid X2 be t h e f i r s t and t h e second p r o j e c t i o n s of X , and form u = v X1 and v = 'd X2 i n I .
v
If
u < v , then v E J , and v X is t h e l e a s t element o f y = v l i n e whose f i r s t coordinate is 2 u.
If
u
D(
m
1
v , then
-
{yo,
v yil
X
=
yu
if
uw v
ciu
if
u
is m-generated
To prove t h i s , one generates t h e four C,
-
{y,,
yi].
ao,
Po,
6
X2. <0, 1 > , < 1 , 0>, a { , p i .
m-generators of
Then t h e four elements of
on t h e
u L X2;
and
= v and u
by
A
A u B
Co
-
{yo,
ydl
and
of o r d e r 1 a r e generated.
One then m-generates C l I 2 - {y1,2, y11/21 2 L. W e alternately m-generate elements of A u B and Ct {yt, y;]. A t each s t e p , we increment t h e order
-
115
Some free m-lattices on posets
t
V(at
yi
t.
A u B, and the order of
of elements of
t
J
and
t
Finally, for any nonzero
In the remainder of t h i s section, we discuss m). An element a = < r , s> of A i s join reducible) i n C ( m ) i f f o r d ( r ) > o r d ( s ) . (Such lower cover a*, and a, 2 x whenever a > x i n element of A i s doubly m-reducible i n D( m ) .
x
Ct,
E
x = t
6
x x
E
x
E
yt
t
E
and
E
I,
i).
<
Ij(
If
i
some further properties of reducible (completely join an element a has a unique D( m).) I n particular, no
I , then x ism-join reducible i n t f 0 ; or
D( m )
J and x = eo or x E [ a i , b i , ei] for 0 is a join reducible element of C ( , t o ) ; or
a, or x = bm; or r c i , d i , f i ) for i
<
<
iff
i
m.
I n verifying that each of the above elements is m -join reducible, most cases If x is an element of Ct t h a t we have not l i s t e d , follow from Figure 3.
x
y i n C t : moreover, d < x implies d 5 y i n D ( m 1. Thus, x is completely join irreducible, and -a f o r t i o r i x is m-join irreducible. We leave the exceptional cases x y o and x = f t o the reader. The formulas for the m-meet reducible elements are m similar, One consequence of t h i s analysis is t h a t C( m ) contains no doubly m-reducible element. then, w i t h two exceptions,
4.
has a unique lower cover
JUSTIFYING THE DESCRIPTION OF
D'
Let
D' is
that
-
D( m )
denote
D( m )
F,(H).
Certainly, C '
[yo
yi).
We s h a l l comment on our two proofs
contains
v
H
as a subposet, and
t1 m -gen
Moreover, for any x E H, x # ( y E H I x # y ) i n D'. By the erates D ' . Structure Theorem of Crawley and Dean [21 for free m-lattices on posets, it only remains t o show that D' s a t i s f i e s the condition (W,). An m-lattice L s a t i s f i e s (Wm) i f f whenever S and T are nonempty m-subsets of L ( I S I ,
IT1 some
m
<
t
1, then /\S sV T
E
implies s
5
VT
for sane
s c S,
or A S s t
for
T.
Let u s c a l l an interval
[a, bl
of
D' a
(W )-failure i f there are non-
m -
S , T of D ' such that a = I \ S , b = V T , but S n [ a , b l empty m-subsets and T n [ a , b l are both empty. One way t o show that D ' is F,(H) is t o prove that D' does not contain any (W,)-failures. We have done t h i s i n [91. For example, suppose there is a
(Wm)-failure
[a, bl
w i t h both
a
and
b in
A. Let x be the greatest element on the x-line through a , and l e t y be the l e a s t element on the y-line through b. Since some element of S must be on the x-line through a, x L b. Consequently, n 2 ( x ) > n2(b) = n 2 ( y ) .
Similarly,
n,(y)
<
n,(a) = n , ( x ) .
Since t h i s means t h a t the shadows of
x
G. Gratzer, D.Kelly
116
and
y
o v e r l a p , such a
also satisfy or
b
in
(Wm). C
(WT)-failure is impossible.
Of course,
C( rn )
niust
I n f a c t , our proof t h a t t h e r e is no (W ) - f a i l u r e with a M C( m ) s a t i s f i e s (h). The A and b E B , required t h e most complicated argument.
is e s s e n t i a l l y an an al y s i s of why
f i n a l case, t hat
a
E
For our second proof i n [ l o ] , we assumed t h e de sc ription of F(H) given by I . Rival and R. Wille [16]. We then developed some elementary universal algeb r a ic lemmas t o show t h a t D' is Frn(H). The S t r u c t u r e Theorem was not used but
we did r e q u i r e t h e f a c t t h a t
C(
rn)
-{y,
advantage of t h e f a c t t h a t t h e elements of f r e e l y rn-generate C {yo, y i l .
-
+ 2). This proof ta ke s labeled with a, a', 6, and p '
is
y'}
C
Fm(Z
Some free m-lattices on posets
117
Proof: -
We s h a l l p r o v e i t b y i n d u c t i o n on n = ( H I + ( K ( t h a t c o n d i t i o n s ( i v )
and ( v ) i m p l y t h e c o n d i t i o n ( i i ) o f theorem 3. L e t us now c o n s i d e r H,,H, K 1 /L K 2 ,
c'f
and K,,K,
e
f,such
that H15H2.
I K ~ ( + ( H (I = n , f ( H l ) = f ( H 2 ) and f ( K l ) = f ( K 2 ) .
a) I f K l i s n o t a s t a b l e , t h e n r i s i n q c o n d i t i o n ( i v )
,( H I I + I K l l < n
and t h u s b y i n d u c t i o n we have f i n i s h e d .
b ) If K l i s a s t a b l e . KI
L e t us c o n s i d e r G1=H1 al.
I f e v e r y q e n e r a l i z e d component o f
Comp(Gl)
then
is
a
stable,
transitive orientations.
(
Comp(G1)
admits
only
two
different
S H E V R I N and FILIPPOV have f i r s t n o t i c e d
t h i s f a c t i n 1181). And t h e n t h e r e s u l t i s g i v e n b y c o n d i t i o n ( i ) . E l s e Comp(G1) a d m i t s a q e n e r a l i z e d p r i m e component T w h i c h i s n o t a s t a b l e and t h e r e f o r e : HI(T)
GI =
H'1 bl
Since (H'I(
9
KI
, a1
KI = H ' i a1
Hl(T)
7
,bl
+ (Sf(HI(T))I
we can use t h e i n d u c t i o n h y p o t h e s i s .
a
103
THE CONSTRUCTION OF SOMF FREE m-LATTICES ON POSETS George G G T Z E R and David KELLY Department of Mathematics University of Manitoba W i nn i peg, Man i t o ba CANADA Dedicated to E . Corominas
11 est bien connu que les treillis peuvent letre considgrgs, soit comme d e s ensembles ordonnes, soit comme des algsbres universelles. L'approche alg6brique
garantit l'existence d'un treillis libre sur tout ensemble ordonn6 ; c'est-2-dire du treillis engendr6 "le plus librement possible" B partir de l'ensemble ordonn6. Beaucoup des ensembles ordonn6s qui apparaissent naturellement dans les differrntes branches des math6matiques ant une structure de treillis. En fait ce sont habituellement des treillis complets. Cependant, quelques techniques alg6briques ne s'appliquent plus directement aux treillis complets. En particulier,
il n'y a pas de treillis complet libre sur un ensemble B trois 6lQments. Le concept
B la th6orie de l'ordre, tandis que la notion de treillis
de complgtude appartient
libre sur un ensemble ordonne provient de l'approche alggbrique. L'inexisterice de treillis complets libres montre qu'on doit-gtre prudent en essayant de combiner ces deux concepts. Nous rgsolvons cette difficult6 en affaiblissant le concept de compl6tude. un cardinal r6gulier infini. Un treillis m-complet ou m-treillis
Soit m
est un ensemble ordonnQ dans lequel toute partie non vide ayant mains de m.El6ments possede une borne supgrieure et une borne infgrieure. Chaque concept de la thgorie des treillis s'dtend en un concept correspondant dans la th6orie des m-treillis. On omet
m
quand
m
=
K,
car c'est de la th6orie des treillis. I1 est clair
qu'un treillis complet est m-complet pour tout m que l a cardinalit6 d ' u n ensemble ordonnh existe dans
P
. De
plus, si m
P , alors toute borne supgrieure qui
est la borne sup6rieure d'une partie non vide de
dinalit6 est inf6rieure 2
m
est plus grand
P dont la car-
; en bref toute borne sup6rieure est une m-borne
sup6rieure. Cependant, i l existe des m-treillis qui sont complets et dans lesquels toute borne sup6rieure n'est pas une m-borne supgrieure. Soulignons que n o u s prenons uniquement les bornes sup6rieures et inf6rieures d e parties non vides. Bien que les m-treillis aient des opgrations infinitaires, le nombre d e ces opgrations e s t born6 de sorte que les concepts et les mdthodes de l'alggbre universelle s'appliquent. Le point important est qu'il existe un ensemble (non une classe) qui indexe simultan6ment toutes les op6rations
G. Gratrer, D. Kelly
104
bornes superieures et inf6rieures dans chaque m-treillis. Une cons6quence particuliGre est l'existence, p o u r tout ensemble ordonne
P
de
Fm(P),
le
P. F,(P) est l e m-treillis qui contient P comme S O U S m-treillis libre sur ensemble ordonne pour lequel les m-bornes sup6rieures et m-bornes infcrieures sont formees "aussi librement clue possible". (En particulier, ). Plus
P
dr6 par
application isotone d e longement B
F (P)
e s t m-engen-
Fm(P)
F ( P ) est l'unique m-treillis pour lequel toute m dans n'irnporte quel m-treillis L a un unique pro-
prgcisgment, P
qui est un
m-homomorphisme.
Un ensemble ordonne est m r n s'il ne contient aucun d e s trois ensembles I 2 3 1 5 + + ; I , "v + 2. , 'L + 2. . I. RIVAL et R. WILLE ont dGcrit 'L le treillis libre sur tout ensemble ordonne mince. Nous avons Etendu ce resultat
ordonnes suivants
~
F (P) p o u r tout ensemble ordonn6 P . Comme d a m m l'article de I. RIVAL et R. WILLE l'argument cl6 dans la description est la d e s -
aux m-treillis en d6crivant
cription du m-treillis libre sur l'ensemble o r d o n n e
< a2 < a l ,
definie par .a denombrable, m
=
He)
.
F
m
bo
< b2 < b , ,
.a
H = {ao,a, ,a2,bo,bl,b2}
< bl
bo
< at .
Pour
m
non
(H) est un treillis complet (ce qui n'est p l u s vrai lorsque
Dans cet article de synthsse, n o u s decrivons F (H) m
et
Fm(H)
.
C o m e le m-treillis
est assez compliqu6, il ne peut pas stre appr6hend6 immgdiatement. Nous
donnons beaucoup de diagrammes pour expliquer l e s formules donnant les bornes
sup6rieures et inferieures dans
F,(H)
.
Une motivation intuitive est aussi donnee
p o u r beaucoup de ces formules. La preuve que le rn-treillis dgcrit est
Fm(H)
est
publi6e ailleurs.
Abstract
Let an
rn be an i n f i n i t e regular c a r d i n a l . We c a l l a Loset m-ccnplete l a t t i c e (or m l a t t i c e ) i f every nortenipty
m elements has both a j o i n and a subset with fewer thsn nieet. This concebt was introduceu by P. Crawley and h.A. Lean. For m - l a t t i c e s , enough of the a l g e b r a i c f l a v o r of the usual ( f i n i t a r y ) l a t t i c e case has been retained so t h a t Lie there are f r ee m - l a t t i c e s generated by any poset. s h a l l d e s c r i b e the f r e e m l a t t i c e on a p a r t i c u l a r 6-element poset 11, and i n d i c a t e w h y t h e d e s c r i p t i o n is c o r r e c t . l'he poset ii is a n a t u r a l example because i t s f r e e m-latt i c e can be used t o describe t h e f w e m - l a t t i c e on many other posets. IHlHOCUCTIOII
I t is well known t h a t l a t t i c e s can be considered e i t h e r a s ordered s e t s or a s universal algebras. The a l g e b r a i c approach guarantees the e x i s t e n c e of t h e f r e e l a t t i c e on any p s e t ; t h a t i s , the l a t t i c e t h a t is kenerated "as f r e e l y a s possibletr s t a r t i n g from t h e poset. ( I f no ordering is specif'ied on t h e gerlera t i n g p o s e t , 2n antichain is undtrstood. )
105
Some free m-lattices on posets
Eiany of t h e ordered s e t s t h a t n a t u r a l l y occur i n various f i e l d s of n:stkienl&t i c s have a l a t t i c e s t r u c t u r e . I n f a c t , they a r e usually com.~lete l a t t i c e s . llowever, sorile e l g ~ b r a i ctechniques no l o n t e r apply d i r e c t l y t o complete l a t t i c e s . I n p a r t i c u l a r , t h e r e is no f r e e complete l a t t i c e on t h r e e elen,ents. If such a complete l a t t i c e d i d e x i s t , then every coicblete l a t t i c e L w i t h t h r e e g e n e r a t o r s would be a complete homoniorpliic image of i t ; i n p a r t i c u l a r , t h e r e would be ari L. llowever, E'iLure 1 shows a upper bound f o r t h e c a r d i n a l i t y of any such 3-generated cornblete l a t t i c e of any given c a r d i n a l i t y . (In t h i s f i t u r e , the t h r e e conlplete generators a r e denoted by squares. ) O f c o u r s e , any diagrem of an i n f i n i t e poset r e q u i r e s i r i t e r p r e t a t i o n . Ari extreme p o s i t i o n is t h a t any such diaLran, is u s e l e s s . h'e a r e not of' t h i s opinion, b u t we do advise extreine c a r e , e s p e c i a l l y i n a proof, when r e a d i n t information fron, a ciisgran; of a i i n f i n i t e poset. b'or example, it is not obvious when an element is j o i n i r r e d u c i b l e . (F'or a f i n i t e l a t t i c e , j o i n i r r e d u c i b l e elements a r e those w i t h a unique lower cover.)
The left-hand chain C i n Figure 1 can be ariy well-orderec chain t h s t bias a l a r b e s t element, denoted by t . The r i ~ h t - h a n d choin is isomorphic t o C , arid t h e r e m i n d e r of t h e poset is a l i n e a r sunl of f i n i t e l s t t i c e s over C . For eacki i t i n C , t h e poset of Figure 1 tias an i-th l e v e l c o n s i s t i n g of' 6 elenients. i < t can only be generatea a f t e r , ari Observe t h a t an elenerit on any l e v e l element on every lower l e v e l has beeti t e n e r a t e d . The concebt of completeness belongs t o order theory. while f r e e l a t t i c e s or1
FIGURE 1 . An a r b i t r a r i l y l a r g e c o m p l e t e l a t t i c e
POSetS come from t h e a l g e b r a i c approsch.
The above example shows t h a t
One
must
G. Gratzer, D. Kelly
106
be c a r e f u l when attempting t o combine these two concepts. d i f f i c u l t y by weakening t h e concept o f completeness.
We s h a l l resolve this
We s h a l l always use t h e symbol m t o denote an i n f i n i t e r e t u l a r c a r d i n a l . An m-complete l a t t i c e , o r i i - l a t t i c e , is a poset i n which every nonempty s u b s e t with fewer than m elements has both a j o i n and a meet. We s h a l l pre fix each l a t t i c e concept with m t o denote t h e Corresponding concept f o r n - l a t t i c e s . Ue m i t m Some examples a r e in-join, ll.-hornornorphism and ni - s u b l a t t i c e . when m :to because t h i s i s t h e usual l a t t i c e case. Obviously, any complete l a t t i c e is an .i-lattice for any ( n . Moreover, P , then any join t h a t e x i s t s i n P i s a j o i n of a nonempty subset of P whose c a r d i n a l i t y is less than .it; i n s h o r t , However, t h e r e a r e it-lattices t h a t a r e complete in any jo i n is an m-join. which not every j o i n is an m j o i n . I f the left-hand chain C i n Figure 1 is t h e ( o r d i n a l ) successor of t h e i n i t i a l o r d i n al corresponding t o m, then t h i s f ig u r e provides such an example L. I n this cas e , note t h a t t h e top element t of C cannot be w r i t t en a s V S f o r a nonempty subset S of L with t L S and IS1 < m. In o t h er words, t is an m-join i r r e d u c i b l e eleuient of L. Consequently, t h e t h r e e squares i n Figure 1 do not m-generate L. if
n exceeds t h e c a r d i n a l i t y of a poset
We emphasize t h a t we only t ak e j o i n s and meets of nonempty s e t s . (In a complete l a t t i c e L , t h e j o i n of t h e empty s e t equals t h e meet of L.) Although m - la tti c e s have i n f i n i t a r y o p er at i o n s , t h er e is a bound on t h e number of such operations so t h a t t h e concepts and methods of universal algebra apply. The important point here is t h a t t h er e is a (not a c l a s s ) t h a t can simultaneously A particular index a l l t h e j o i n and meet o p er at i o n s of every m - l a t t i c e . consequence is t h e ex i s t en ce, f o r any poset P, of F,(P), t h e free m - l a t t i c e
set
on
P.
Fm(P)
is t h e
m-lattice
t h a t co n t ai ns
P
a s a subposet f o r which
m-joins and m-meets a r e formed "as f r e e l y a s possible". (In particular, F,(P) is al-generated by P . ) More p r eci s el y , F A P ) is t h e unique m - l a t t i c e f o r which every isotone map from P t o any m-la ttic e L has a unique extension t o an m-homomorphism from F AP ) t o L. W e s h a l l use n_ , a boldface number, t o denote an n-element chain. A l i g h t f a c e number i n d i c a t e s an a n t i chain. A s u s u a l , P + Q denotes t h e d i s j o i n t union of t h e posets P and Q. We s h a l l c a l l a poset slender i f it does not contain any of t h e following th r e e p o s e t s a s a subposet: (i)
(ii) (iii)
3=1+1+1 2, + 1+
2 5
.
I. Rival and R. Wille [ I 6 1 described t h e f r e e l a t t i c e on any f i n i t e slender poset. I n a sequence o f papers "91, [ l o ] , [ill, [121), we have generalized t h i s r e s u l t by describing F-(P) f o r any slender poset. Our techniques a l s o work when m is countable, and so provide another proof of t h e Rival-Wille r e s u l t . However, since t h e rn A case d i f f e r s s i g n i f i c a n t l y from a l l o t h e r s , we
henceforth assume t h a t
ni
is uncountable.
S i m i l a r l y a s i n [16], t h e key ingredient i n describing t h e free m - l a t t i c e on any slender poset is a d es cr i p t i o n of t h e f r e e m -la ttic e on t h e poset H of Figure 2. The present paper is devoted t o describing Fm(H). We s h a l l se e t h a t
is a complete l a t t i c e (which would not be t r u e i f we allowed m F,(H) In g e n e r a l , we s h a l l only i n d i c a t e proofs. W e not only de sc ribe F,(R)
No). as a
Some free m-lattices on posets
107
p o s e t , but we L i v e t h e formulas !'or ( a r b i t r a r y ncnenpty) j o i n s and meets.
FIGURt 2. The Loset
I n order t o d e s c r i b e
11
Fm(fl), we mst f i r s t introduce sonle o t h e r
n-lat
t i c e s . I n Section 1 , we describe C ( m ) , t h e f r e e m - l a t t i c e on 2 + 2 , w i t h a new 0 and 1. We a l s o r e q u i r e t h e m - l z t t i c e s A and b oescribed i n Section 2. I n Section j , we d e s c r i b e D(m 1, which is Fm(H) with a new C, and 1 . Our paper [91 contains ttie proof's of a l l s t a t e n e n t s nade i r i t h e f'irst three sections. I n Section 4 , we d i s c u s s the p-oofs t h a t t h e m-lattices C ( m ) and C( m ) , a s d e s c r i b e d , a r e what we claimed. One approach w i l l be t o apply t h e S t r u c t u r e Theorem f o r f r e e m - l a t t i c e s on posets given by P. Crawley and Fi.A. Dean [21. Complete d e t a i l s f o r Section 4 can be found i n L91 x i a [ l o ] . Let P be a slender poset. I f F is a l s o l i n e a r l y indecoriiposable, then we F is countable, and t h a t Fm(P) can be embedded, as s n show i n [ill t h a t m-sublattice,
into
does not contain
F,,,(H).
Consequently,
a poset
F is slender i f f
Fm(P)
Fm(3) a s an m-sublattice.
W e conclude t h i s introduction by b r i e f l y mentioninb some f u r t h e r t o p i c s about m-lattices. Free m - l a t t i c e s , defined a s f r e e m - l a t t i c e s over a n t i c h a i n s , can a l s o be considerea a s f r e e ?.-products of one-element m-lattices. For f r e e products, we r e f e r t o G r a t z e r , Lakser and P l a t t [ I 4 1 arid Gratzer and Lakser [ l 3 ] . I n our paper 171, we Generalized sonJe o f t h e r e s u l t s o f ' t h e s e two papers, and a l s o studied what happens when m is v a r i e d . For example, we proved t h e S t u c t u r e Theorem f o r f r e e m-products, which is s i m i l a r t o t h e S t r u c t u r e Theorem f o r f r e e m - l a t t i c e s of Crawley and Dean [21. Another r e s u l t o f [ T I is: I f a covers b i n some f r e e m -product, then a continues t o cover b when m is increased. (Covers a r e of interest becsuse A. Cay [31 has shown t h a t every proper closed i n t e r v a l of a f i n i t e l y generated f r e e l a t t i c e contains a covering p a i r . ) We a l s o determined which f r e e m-products a r e complete i n [TI. I n p a r t i c u l a r , t h e f r e e " - l a t t i c e on t h r e e g e n e r a t o r s i s not complete ( s e e Whitnian [1&1 f o r t h e m = case). When r e p r e s e n t i n g an element of an m l a t t i c e i n terms of an m -generatink set (using m-Joins and m-meets), t h e r e is no general way t o s p e c i f y a unique r e p r e s e n t a t i o n , even i n a f r e e m - l a t t i c e . For example, t h e r e a r e many exaniples in
Fm(3)
of an
m-join
reducible element t h a t cannot be expressed a s an
irredundant m-join. In 171, we generalized t h e Normal Form Theorern t h a t 6 . Jdnsson [151 proved f o r t h e f r e e m - l a t t i c e on a poset. We used t h e t h e norm61 r e p r e s e n t a t i o n o f [7] t o g e n e r a l i z e a r e s u l t about f r e e products i n 1131.
A n a t u r a l question about an ordered s e t is: M a t is t h e l a r g e s t chain t h a t
it c o n t a i n s ?
We say t h a t a poset P c o n t a i n s no chain of c a r d i n a l i t y n . r e g u l a r . M.E. Adams and 0 . K e l l y [ l ]
s a t i s f i e s the n-chain condition i f it Henceforth, l e t n 6 e uncountable and showed t h a t , i f t h e poset P s a t i s f i e s
G. Gratzer, D. Kelly
108
the n-chain c o n d i t i o n , then so does the f r e e l a t t i c e on P. However, f o r m, we contructed, i n our paper [61 with A . llajnal (for any uncountable n > m ) , a poset P with no uncountable chains such t h a t t h e f r e e m - l a t t l c e on P contains a chain of c a r d i n a l i t y n . I n Adam and k e l l y [ l ] , it was a l s o show t h a t t h e f r e e product construction preserves t h e n -chain condition. In our paper [61 with A. Hajnal, we a l s o showed t h a t a f r e e m - l a t t i c e cannot contain a chain with more than m elements. ( I n t h e previous sentence, we assumed t h e Generalized Continuum Hypothesis t o simplify t h e formulation.) If m gl and n is t h e successor c a r d i n a l t o t h e continuum, then we do n'ot know whether the f r e e
1.
m-product construction preserves t h e n-chain c o n a i t i o n .
THE COMPLETE LATTICE
C( m)
Recall t h a t m always denotes an uncountable r e g u l a r c a r d i n a l . The complete l a t t i c e C ( m ) of Figure 3 is t h e f r e e m-lattice on 2, + 2 with a new 0
and
1.
F(2 +
The diagram is a l s o v a l i d f o r
2)
( s e e Rolf [Yl or Chapter VI
of Gratzer [51). W e have denoted t h e 16-element l a t t i c e i n t h e middle of Figure 3 by C( A o ) . (The way t h i s f i n i t e l a t t i c e was used by I. Rival and R. Wille [I61 t o c o n s t r u c t F(H) is s i m i l a r t o t h e way we s h a l l use C ( m ) F A H ) .I I n t h i s s e c t i o n , we only d i s c u s s t h e complete l a t t i c e c( m). We leave u n t i l Section 4 t h e question whether C ( m ) is claimed. Observe t h a t a,, a;, b,, and b h generate C ( K O ) . t h e notation f, a A b and f ' a ' v bh. n m m m
t o construct s t r u c t u r e of what we have W e a l s o use
For every successor o r d i n a l j < in, t h e r e is a lower j-th l e v e l of 6 elements L. l a . , b . , c ., d ., e j , f j ) , and f o r every l i m i t o r d i n a l i < M J J J J J (including i 01, t h e r e is a lower i-th l e v e l of 7 elements Li = { a i , b i , c i , d i , e i , f i , g i ] . These elements a r e ordered a s shown i n Figure 3. There i s a l s o an upper i-th level
f o r each
Ui
i
m, defined d u a l l y and denoted by t h e safile
<
l e t t e r s with primes. For convenience, we a l s o l a b e l 6 elements of ah, p bO, p' bb, y go, y ' = gb. Greek l e t t e r s : a I ao, a '
I n a more formal approach, one d e f i n e s
c( so) uu ( LJ. I
j
C( m
t o be
LI ( uJ.
I j
C(
m)
with
where each f i n i t e poset t h a t we have l i s t e d i s assumed t o be a s u b p s e t of C( T ) . Let i and j be o r d i n a l s l e s s than m The a d d i t i o n a l comparability u < v holds i n C ( m ) e x a c t l y when one of t h e following conditions is s a t i s f ied :
.
u = a i , v = a . and i < j ; J' u = bi, v = b . , and i < j ; J u E Li [ a i , b i l , v E L . , and i J u = ai and a, 5 v i n C ( % o ) ;
-
u
= b.1 and b,
-
5
v
in
C( E'o)
<
j ;
;
E Li { a i , bil and v E C ( g o ) or v E :U J' a previous condition holds w i t h u and v interchanged, L and U interchanged, primed elements, and r e v e r s a l of t h e i n e q u a l i t y i n
u
C(
so);
Some free m-lattices on posets
109
i limit
a:
Ago= y
FIGURE 3.
The complete lattice
C( m 1
G. Gratzer, D. Kelly
110
u
and
*
v
b!; J u = bi and v E U . but v :a!. J J' Clearly, y go is t h e least element of C ( m ) , and y' = pi, is t h e is a complete l a t t i c e . To sketch t h e proof o f t h i s g r e a t e s t . In f a c t , C ( " ) s t a t e m e n t , l e t u s e x p l a i n how t o c a l c u l a t e (nonempty) j o i n s . From Figure 3, one sees a n a t u r a l way to break C ( m ) up i n t o 7 p a r t s : C ( AO), the bottom l e f t chain [ a i I i 5 m I , t h e bottom middle p a r t from go t o k , t h e bottom a.
1
v
U.
E
J'
but
r i g h t chain bi j i 5 m I , and t h e corresponding t h r e e t o p p a r t s . C( has e x a c t l y one p o i n t i n common with each of t h e o t h e r six p a r t s . Observe t h a t C( m ) . each p a r t is a complete l a t t i c e , i n f a c t , a complete s u b l a t t i c e o f Since each i n f i n i t e p a r t is a l i n e a r sum over a complete c h a i n , it should be c l e a r how t o c a l c u l a t e t h e j o i n i n each p a r t . I t only remains t o determine t h e W e leave the j o i n o f p a i r s of incomparable elements frorn d i f f e r e n t p a r t s . d e r i v a t i o n of t h e s e formulas t o t h e r e a d e r .
8'.
Let u s i n d i c a t e why C( m ) We f i r s t g e n e r a t e elements of
-
[y, y ' ) is m-generated by a , a ' , C ( &o) i n t h e following o r d e r :
since 0 i d e a f o r t h e rest of t h e argument complete l a t t i c e of Figure 1. I f i < m , then it is easy t o generate
We can now generate
a l i m i t ordinal then 2.
a. J
and
C( X )
j
Li
m,
<
we have its four g e n e r a t o r s ,
.
p, and
We kave t h e
i n t h e i n t r o d u c t i o n when we discussed t h e Li has a l r e a d y been m-generated f o r some Li+l using f i n i t e j o i n s and meets. I f , for
has already been =-generated
b . can then be m-generated s i n c e J
a. =
( ai
J
f o r each
I i
<
j)
.
i
<
j,
THE LATTICE A
I n t h i s section, we d e f i n e t h e second b u i l d i n g block of D( m ) , t h e l a t t i c e of F i g u r e 4. This l a t t i c e was discovered by I . Rival and R . Wille L161. Let J be t h e set of dyadic r a t i o n a l s r t h a t s a t i s f y 0 5 r s 1. Every nonzero r E J has a unique r e p r e s e n t a t i o n , t h e normal form, r = a * 2-", where a is an odd
A
integer;
n
is t h e
order of
r; i n notation, n
= 0. Observe t h a t o r d ( r ) = o r d ( 1 maximum of o r d ( r ) and o r d ( s ) . We d e f i n e
A
a s a subposet of
- r). J2
Although
< r , s>
E
A , then
v
o r d < r , s> t o b e t h e
-ord
not
with t h e exception of J 2 , For example,
a s u b l a t t i c e of
<1/4, 1/2> = <1/2, I > A
can be redefined i n two ways:
A = I
<
s and
s
[ < r , s> I r
<
s
and
s
A
W e a l s o define
o r d ( r ) ,i ord(s)
is a l a t t i c e , it is
A
Ey convention, o r d ( 0 )
with t h e product o r d e r :
A = { < r , s > 1 r < sand Note t h a t i f
ord(r).
-
r = 2-",
n >. o r d ( r ) l
r = 2-",
n
2
ord(s)}
.
<0, l > .
111
Some free m-lattices on posets
We s h a l l use
and
nl
t o denote t h e f i r s t and second p r o j e c t i o n s of
n2
J2
J . For t E J , l e t us c a l l t h e set of a E A t h a t s a t i s f y n , ( a ) t x = t l i n e i n A , and d e f i n e t h e y z t l i n e similarly. The a l t e r n a t e d e f i n i t i o n s of A y i e l d e x p r e s s i o n s f o r t h e x = r and y s l i n e s i n A. In > is t h e l a r g e s t element on t h e x = r l i n e , and p a r t i c u l a r , < r , r + 2- Ord(')
onto
the
Each a formulas a r e :
For example, 1>1
denoted by
has a r i g h t lower cover
a*
and a r i g h t upper cover
a*.
The
< ( r + s ) / 2 , s> < r , ( r + s ) / 2 >.
< r , s>* < r , s>, {
A
-
<1/4, 1/2>* <3/8, 1/2> i n Figure 4. Moreover, i f a E A s a t i s f i e s o r d ( a ) = o r d ( n , ( a ) ) , then a has a l e f t lower c o v e r ,
*a, and d e f i n e d by
* < r , s> = < r
-
2-ord(r), s>.
-ord ( s ) > S i m i l a r l y , t h e l e f t upper cover * < r , s> e x i s t s and e q u a l s < r , s + 2 when o r d ( r ) < o r d ( s ) . Observe t h a t <0, 1> has no l e f t (lcwer or upper) c o v e r , b u t t h a t every o t h e r element o f A h a s e x a c t l y one l e f t cover. Moreover, each cover o f an element a E A l i e s on t h e x-line through a or t h e y-line through a .
FIGURE 4.
The l a t t i c e
A
I12
G. Gratzer, D. Kelly
.
2 I n Figure 4 , A seenis t o be a very s p a r s e subset of J For each a = < r , s> E A, l e t us c a l l the closed i n t e r v a l [ r , s ] ( i n e i t h e r t h e r e a l s or dyadics) t h e shadow of a . We say t h a t two closed i n t e r v a l s overlap i f they have an i n t e r v a l of nonzero length i n common, but n e i t h e r i n t e r v a l contains t h e o t h e r . The " r e l a t i v e sparseness'! of A is expressed by t h e following shadow p r i n c i p l e : The shadows of two elements of A cannot overlap. Let u s i n d i c a t e an easy way W e add a v e r t i c a l l i n e ( t h e shadow l i n e ) t h a t passes t o v i s u a l i z e shadows. through t h e i n t e r s e c t i o n of the x = 1/2 and y = 1/2 lines on t h e r i g h t s i d e of A i n Figure 4 ; index t h i s l i n e i n t h e obvious way w i t h 10, 11. I f one places an opaque r i g h t angle a t a E A (forming t h e x - l i n e and t h e y-line through a ) , then t h e i n t e r s e c t i o n of t h e r i g h t angle with t h e shadow l i n e g i v e s t h e shadow of a . W e found t h e shadow p r i n c i p l e t o be a very n a t u r a l t o o l i n our proofs. Let u s apply it t o "explaint1 t h e emptiness of t h e l a r g e s t open square i n Figure 4: i f t h e r e was an element a i n t h i s square, then t h e shadows of a and <0, 1/2> would overlap.
I n Figure 4 , the d i s t a n c e of each element < r , s> E A from t h e shadow l i n e is proportional t o s r , t h e length of its shadow. Consequently, f o r two incomparable elements a , b E A , we can define a t o be t o t h e l e f t of b i f f
-
has a longer shadow than b. Observe t h a t a is t o t h e l e f t of b i f f < n,(b) iff n2(a) > n 2 ( b ) . With t h i s concept, we can v i s u a l i z e t h e 1 f i n i t a r y j o i n i n A. Let a and b be incomparable elements of A , with a t o t h e l e f t of b . The j o i n of a and b is t h e l e a s t element on t h e y - l i n e through a t h a t is g r e a t e r than b. Consequently, a V b an f o r ttie l e a s t n
a
n (a)
such t h a t an > b , where an is inductively defined by: a. T h i s shows t h a t A is a l a t t i c e .
3.
THE COMPLETE LATTICE D(
a , ai+l = ( a i ) *
.
t?t)
W e build up D( m ) from t h r e e d i s j o i n t s u b l a t t i c e s : l a t t i c e A was defined i n Section 2. We d e f i n e B = { < r , s> I < s , r >
E
A , B , and C.
The
A],
a subposet of J2. Clearly, B is a l a t t i c e and its diagram is obtained by r e f l e c t i n g Figure 4 about a v e r t i c a l l i n e . I n o t h e r words, mapping < r , s> E A to
b
of
B
has l e f t covers
a r i g h t lower cover
b*
or
a
*b
and
*b,
and i f
r i g h t upper cover
b z < 1 , 0 > , then
b
b*.
F i n a l l y , l e t I be t h e r e a l i n t e r v a l [O, 11, and r e c a l l t h a t J denotes t h e s u b s e t of I c o n s i s t i n g of dyadic r a t i o n a l s . For each t E J , we t a k e a and generators a t , a ; , pt, copy Ct of C( m ) , w i t h bounds yt and y t ,
p i , For each t E I two-element chain with
t
E
I.
Since
I
which is riot a dyadic r a t i o n a l , Ct = [y,, y i ] is t h e yt < y.; W e define C a s t h e l i n e a r sum of t h e C t ,
is complete and each
Ct
is complete,
C
is a complete
lattice. To d e f i n e C ( m ) , ( s e e Figures 5 and 6):
we must describe t h e p a r t i a l ordering on
A u b u C
Some free m-lattices on posets
FIGURE 5.
The complete l a t t i c e
FIGURE 6.
Cetails of
fi( m )
113
D(
rn)
G. Gratzer, D. Kelly
114
If
S>
E
< r , s> < r , s> < r , s> < r , s>
< t ,u> < t , u>
A, <
> <
> <
>
< t , u> E B, v E I , and < t , u> i f f s < u; < t , u> i f f r > t ; p i f f s < v holds o r s p i f f r > v holds o r r p i f f t < v holds or t p i f f u > v holds o r u
I t is e a s i l y seen t h a t
Cv,
then:
v v
and
av
and
v
and and
a' 2 V
p
= = = =
E
v
is a poset.
D(m)
_c
pv p{
5
p
hold;
p
hold;
p
hold; hold.
c
2
However, since <0, 1/2> 1 < 1 , 1 / 2 > , v was assumed t o be a dyadic pv, o r p.; Tnus, i f v is not
2.
A u B is not a subposet of J Note t h a t r a t i o n a l wherever we used t h e notation a v , a ; ,
dyadic, then i n each of the l a s t four c a s e s , only t h e f i r s t c l a u s e can apply.
I t is not d i f f i c u l t t o show t h a t C( m ) is a l a t t i c e , and t h a t each of A , B, and C is a s u b l a t t i c e of D( m ) . For < r , s> E A , < t , u> E B , v E I , and p E C v , we give t h e formulas f o r joining p a i r s :
< r , s> v p
(a)
( i ) as v p
is E
(ii) < r , s>,
(b)
C , where t h e j o i n is formed i n
if
r
>
v
r
or
v
w
and
(iii)
t h e l e a s t
(iv)
t h e l e a s t
p
C, i f
r r
5 5
s
5
v v
<
s and
< s
and
< r , s> v < t , u> i s ( i ) < t , u>, i f s < u; ( i i ) < r , s>, i f t < r ; ( i i i ) the l e a s t
E
@
pt
of
(a;
D( m )
and p i )
v;
i n Cv;
5 a;
t h a t maps f o r any
p .k a; i n Cv; p 5 i n Cv;
> >
t , i f s >t; s, if s >t;
< r , s> E A t o t E J . In o t h e r
words, $ r e f l e c t s Figure 5 a b u t i t s c e n t r a l a x i s . With t h i s observation, t h e formulas f o r < t , u> v p follow from ( a ) . The pairwise meet formulas follow by duality. In order t o show t h a t D( m ) is a complete l a t t i c e , it s u f f i c e s t o f i n d X f o r any nonempty subset X of A. (The formula is s i m i l a r f o r B and we already know t h a t C is complete.) Let X, arid X2 be t h e f i r s t and t h e second p r o j e c t i o n s of X , and form u = v X1 and v = 'd X2 i n I .
v
If
u < v , then v E J , and v X is t h e l e a s t element o f y = v l i n e whose f i r s t coordinate is 2 u.
If
u
D(
m
1
v , then
-
{yo,
v yil
X
=
yu
if
uw v
ciu
if
u
is m-generated
To prove t h i s , one generates t h e four C,
-
{y,,
yi].
ao,
Po,
6
X2. <0, 1 > , < 1 , 0>, a { , p i .
m-generators of
Then t h e four elements of
on t h e
u L X2;
and
= v and u
by
A
A u B
Co
-
{yo,
ydl
and
of o r d e r 1 a r e generated.
One then m-generates C l I 2 - {y1,2, y11/21 2 L. W e alternately m-generate elements of A u B and Ct {yt, y;]. A t each s t e p , we increment t h e order
-
115
Some free m-lattices on posets
t
V(at
yi
t.
A u B, and the order of
of elements of
t
J
and
t
Finally, for any nonzero
In the remainder of t h i s section, we discuss m). An element a = < r , s> of A i s join reducible) i n C ( m ) i f f o r d ( r ) > o r d ( s ) . (Such lower cover a*, and a, 2 x whenever a > x i n element of A i s doubly m-reducible i n D( m ) .
x
Ct,
E
x = t
6
x x
E
x
E
yt
t
E
and
E
I,
i).
<
Ij(
If
i
some further properties of reducible (completely join an element a has a unique D( m).) I n particular, no
I , then x ism-join reducible i n t f 0 ; or
D( m )
J and x = eo or x E [ a i , b i , ei] for 0 is a join reducible element of C ( , t o ) ; or
a, or x = bm; or r c i , d i , f i ) for i
<
<
iff
i
m.
I n verifying that each of the above elements is m -join reducible, most cases If x is an element of Ct t h a t we have not l i s t e d , follow from Figure 3.
x
y i n C t : moreover, d < x implies d 5 y i n D ( m 1. Thus, x is completely join irreducible, and -a f o r t i o r i x is m-join irreducible. We leave the exceptional cases x y o and x = f t o the reader. The formulas for the m-meet reducible elements are m similar, One consequence of t h i s analysis is t h a t C( m ) contains no doubly m-reducible element. then, w i t h two exceptions,
4.
has a unique lower cover
JUSTIFYING THE DESCRIPTION OF
D'
Let
D' is
that
-
D( m )
denote
D( m )
F,(H).
Certainly, C '
[yo
yi).
We s h a l l comment on our two proofs
contains
v
H
as a subposet, and
t1 m -gen
Moreover, for any x E H, x # ( y E H I x # y ) i n D'. By the erates D ' . Structure Theorem of Crawley and Dean [21 for free m-lattices on posets, it only remains t o show that D' s a t i s f i e s the condition (W,). An m-lattice L s a t i s f i e s (Wm) i f f whenever S and T are nonempty m-subsets of L ( I S I ,
IT1 some
m
<
t
1, then /\S sV T
E
implies s
5
VT
for sane
s c S,
or A S s t
for
T.
Let u s c a l l an interval
[a, bl
of
D' a
(W )-failure i f there are non-
m -
S , T of D ' such that a = I \ S , b = V T , but S n [ a , b l empty m-subsets and T n [ a , b l are both empty. One way t o show that D ' is F,(H) is t o prove that D' does not contain any (W,)-failures. We have done t h i s i n [91. For example, suppose there is a
(Wm)-failure
[a, bl
w i t h both
a
and
b in
A. Let x be the greatest element on the x-line through a , and l e t y be the l e a s t element on the y-line through b. Since some element of S must be on the x-line through a, x L b. Consequently, n 2 ( x ) > n2(b) = n 2 ( y ) .
Similarly,
n,(y)
<
n,(a) = n , ( x ) .
Since t h i s means t h a t the shadows of
x
G. Gratzer, D.Kelly
116
and
y
o v e r l a p , such a
also satisfy or
b
in
(Wm). C
(WT)-failure is impossible.
Of course,
C( rn )
niust
I n f a c t , our proof t h a t t h e r e is no (W ) - f a i l u r e with a M C( m ) s a t i s f i e s (h). The A and b E B , required t h e most complicated argument.
is e s s e n t i a l l y an an al y s i s of why
f i n a l case, t hat
a
E
For our second proof i n [ l o ] , we assumed t h e de sc ription of F(H) given by I . Rival and R. Wille [16]. We then developed some elementary universal algeb r a ic lemmas t o show t h a t D' is Frn(H). The S t r u c t u r e Theorem was not used but
we did r e q u i r e t h e f a c t t h a t
C(
rn)
-{y,
advantage of t h e f a c t t h a t t h e elements of f r e e l y rn-generate C {yo, y i l .
-
+ 2). This proof ta ke s labeled with a, a', 6, and p '
is
y'}
C
Fm(Z
Some free m-lattices on posets
117
Proof: -
We s h a l l p r o v e i t b y i n d u c t i o n on n = ( H I + ( K ( t h a t c o n d i t i o n s ( i v )
and ( v ) i m p l y t h e c o n d i t i o n ( i i ) o f theorem 3. L e t us now c o n s i d e r H,,H, K 1 /L K 2 ,
c'f
and K,,K,
e
f,such
that H15H2.
I K ~ ( + ( H (I = n , f ( H l ) = f ( H 2 ) and f ( K l ) = f ( K 2 ) .
a) I f K l i s n o t a s t a b l e , t h e n r i s i n q c o n d i t i o n ( i v )
,( H I I + I K l l < n
and t h u s b y i n d u c t i o n we have f i n i s h e d .
b ) If K l i s a s t a b l e . KI
L e t us c o n s i d e r G1=H1 al.
I f e v e r y q e n e r a l i z e d component o f
Comp(Gl)
then
is
a
stable,
transitive orientations.
(
Comp(G1)
admits
only
two
different
S H E V R I N and FILIPPOV have f i r s t n o t i c e d
t h i s f a c t i n 1181). And t h e n t h e r e s u l t i s g i v e n b y c o n d i t i o n ( i ) . E l s e Comp(G1) a d m i t s a q e n e r a l i z e d p r i m e component T w h i c h i s n o t a s t a b l e and t h e r e f o r e : HI(T)
GI =
H'1 bl
Since (H'I(
9
KI
, a1
KI = H ' i a1
Hl(T)
7
,bl
+ (Sf(HI(T))I
we can use t h e i n d u c t i o n h y p o t h e s i s .
a