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Pacific Journal <strong>of</strong><br />
Mathematics<br />
CHAIN CONDITIONS IN FREE PRODUCTS OF LATTICES<br />
WITH INFINITARY OPERATIONS<br />
GEORGE GRÄTZER, ANDRAS HAJNAL AND DAVID C. KELLY<br />
Vol. 83, No. 1 March 1979
PACIFIC JOURNAL OF MATHEMATICS<br />
Vol. 83, No. 1, 1979<br />
CHAIN CONDITIONS IN FREE PRODUCTS OF LATTICES<br />
WITH INFINITARY OPERATIONS<br />
G. GRATZER, A. HAJNAL AND DAVID KELLY<br />
There are many facts known about the size <strong>of</strong> subsets <strong>of</strong><br />
certa<strong>in</strong> k<strong>in</strong>ds <strong>in</strong> <strong>free</strong> <strong>lattices</strong> and <strong>free</strong> <strong>products</strong> <strong>of</strong> <strong>lattices</strong>.<br />
Examples: every cha<strong>in</strong> <strong>in</strong> a <strong>free</strong> lattice is at most countable;<br />
every "large" subset conta<strong>in</strong>s an <strong>in</strong>dependent set; if the<br />
<strong>free</strong> product <strong>of</strong> a set <strong>of</strong> <strong>lattices</strong> conta<strong>in</strong>s a "long" cha<strong>in</strong>,<br />
so does the <strong>free</strong> product <strong>of</strong> a f<strong>in</strong>ite subset <strong>of</strong> this set <strong>of</strong><br />
<strong>lattices</strong>. Here we <strong>in</strong>vestigate these problems <strong>in</strong> the sett<strong>in</strong>g<br />
<strong>of</strong> a variety V <strong>of</strong> m-<strong>lattices</strong>, where m is an <strong>in</strong>f<strong>in</strong>ite regular<br />
card<strong>in</strong>al. An m-lattice L is a lattice <strong>in</strong> which for any<br />
nonempty set S <strong>with</strong> |S| m is<br />
the successor <strong>of</strong> a regular card<strong>in</strong>al, then it is strongly m-<strong>in</strong>accessible.<br />
Let π > m be regular and strongly m-<strong>in</strong>accessible. The Erdos-<br />
Rado theorem [3, Lemma 1] states that for any family (Sa \a
108 G. GRATZER, A. HAJNAL AND DAVID KELLY<br />
<strong>of</strong> L <strong>in</strong>to K. In particular, if each Z^ (i e /) is a one-element lattice,<br />
then L is the V-<strong>free</strong> m-lattice generated by X. We omit mention <strong>of</strong> V<br />
if it is the variety L m <strong>of</strong> all m-<strong>lattices</strong>. We also omit m if m = fc$ 0<br />
Let X = {x a \a
CHAIN CONDITIONS IN FREE PRODUCTS OF LATTICES 109<br />
result due to F. Galv<strong>in</strong> and B. Jόnsson [4] <strong>in</strong> the m = ^ 0 case.<br />
COROLLARY 3. Let n be a regular card<strong>in</strong>al that is greater than<br />
m and strongly m-<strong>in</strong>accessible. If a set <strong>of</strong> card<strong>in</strong>ality n is a<br />
subset <strong>of</strong> a <strong>free</strong> m-lattice, then it conta<strong>in</strong>s an m-<strong>in</strong>dependent subset<br />
<strong>of</strong> the same card<strong>in</strong>ality.<br />
B. Jόnsson [9] proved that the F-<strong>free</strong> product <strong>of</strong> <strong>lattices</strong> (Lt\i e<br />
I) satisfies the m-cha<strong>in</strong> condition (m is regular and >V$ 0) iff for all<br />
f<strong>in</strong>ite Γ £ J, the F-f ree product <strong>of</strong> (Lt\ie I f<br />
) satisfies the m-cha<strong>in</strong><br />
condition. Our next result generalizes this.<br />
THEOREM 2. Let V be an m-variety. Let nbe a regular card<strong>in</strong>al<br />
that is greater than m and strongly m-<strong>in</strong>accessible. Let L be the<br />
V-<strong>free</strong> m-product <strong>of</strong> the m-<strong>lattices</strong> L te V, ίel. If r for all J Q I<br />
<strong>with</strong> \J\ < m, the <strong>free</strong> m-product <strong>of</strong> (L t\isJ) satisfies the n-cha<strong>in</strong><br />
condition, then so does L.<br />
If n is s<strong>in</strong>gular and c<strong>of</strong><strong>in</strong>al <strong>with</strong> ^ 0, then there are two <strong>lattices</strong><br />
satisfy<strong>in</strong>g the n-cha<strong>in</strong> condition whose F-<strong>free</strong> product does not<br />
satisfy the n-cha<strong>in</strong> condition. If n is c<strong>of</strong><strong>in</strong>al <strong>with</strong> ^ 0> then there<br />
are countably many cha<strong>in</strong>s <strong>of</strong> card<strong>in</strong>ality m is an <strong>in</strong>f<strong>in</strong>ite card<strong>in</strong>al <strong>of</strong> c<strong>of</strong><strong>in</strong>ality m 0<br />
<strong>with</strong> m 0
110 G. GRATZER, A. HAJNAL AND DAVID KELLY<br />
Hence, we can assume that each element <strong>of</strong> Y has a proper representation<br />
a = p(ά), where the same m-polynomial p is used for each<br />
element <strong>of</strong> Y. For notational simplicity, we further assume that,<br />
for some card<strong>in</strong>al m < m, a — (x 0 a<br />
a \ a < m> whenever ae Y, where<br />
o<br />
tfel for all a < m . (Note that x 0 a<br />
Φ xa for a Φ β.)<br />
a β<br />
Consider the sets S = {x a a<br />
a \ a < m } for a e Γ. By the Erdόs-Rado<br />
0<br />
theorem, there is a subset 3Γ' £ 3f <strong>with</strong> | Y' \ — n such that (Sa\a e<br />
Y f<br />
) is a J-system, whose kernel we denote by D. For each ae Y',<br />
the <strong>in</strong>clusion D Q Sa <strong>in</strong>duces a map >fra: D-+ m0 <strong>in</strong> the obvious way.<br />
S<strong>in</strong>ce \{fa\ae Y'}\
<strong>in</strong> L.<br />
CHAIN CONDITIONS IN FREE PRODUCTS OF LATTICES 111<br />
Pro<strong>of</strong>. Let L b = L U {0, 1}, the tn-lattice formed by add<strong>in</strong>g a<br />
new zero and one to L. It is easily seen that L b e V. Further, let<br />
0 and 1 be the 7-sequences <strong>with</strong> constant entry 0 and 1, respectively.<br />
We are assum<strong>in</strong>g that (i) p(a, c) g q(b, d) <strong>in</strong> L and (ii) p(a, e) = q(b, e)<br />
<strong>in</strong> K. By consider<strong>in</strong>g the m-homomorphism from L to L b that maps<br />
I/ o identically, everyth<strong>in</strong>g <strong>in</strong> Lx to 1, and every<strong>in</strong>g <strong>in</strong> L2 to 0, we<br />
conclude from (i) that p(a, 1) (α, 0)^p(α, c)<br />
and g(6, d) ^ g(6, 1) <strong>in</strong> I, 6<br />
.<br />
desired conclusion.<br />
Therefore, q(b, d) ^ />(α, c) <strong>in</strong> L, the<br />
Let it be as <strong>in</strong> the statement <strong>of</strong> Theorem 2, let L be the F-<strong>free</strong><br />
m-product <strong>of</strong> the family (Lt\ίel) <strong>of</strong> nx-<strong>lattices</strong>, and let X=\J(Li\ie<br />
/), a subset <strong>of</strong> L. Suppose that C is a cha<strong>in</strong> <strong>in</strong> L <strong>of</strong> card<strong>in</strong>ality<br />
tt. As <strong>in</strong> the pro<strong>of</strong> <strong>of</strong> Theorem 1, we can assume that there is a<br />
s<strong>in</strong>gle m-polynomial p and a card<strong>in</strong>al m0 < rπ such that each element<br />
a <strong>of</strong> C has a representation α = p({xl\a < mo», where tfel for<br />
all a < m0. For x e X, i(x) denotes the element j <strong>of</strong> I such that<br />
xeLd. S<strong>in</strong>ce there are less than n equivalence relations on m0, we<br />
can further assume that, whenever a, β < nt0, if the equality i(x%) =<br />
i(xf) holds for some aeC, then it holds for all aeC.<br />
Apply<strong>in</strong>g the Erdos-Rado theorem to the sets Sa = {i(Xa)\oc
112 G. GRATZER, A. HAJNAL AND DAVID KELLY<br />
LEMMA 2. Let n be a strongly m-<strong>in</strong>accessible card<strong>in</strong>al whose<br />
c<strong>of</strong><strong>in</strong>ality is greater than 2~. If (P t\ίel) is a family <strong>of</strong> posets<br />
<strong>with</strong> 0 and 1 satisfy<strong>in</strong>g the n-cha<strong>in</strong> condition, then Z? m(P i|ΐe I)<br />
satisfies the n-cha<strong>in</strong> condition.<br />
Pro<strong>of</strong>. Suppose C is a cha<strong>in</strong> <strong>in</strong> i7m(Pΐ | i e I) <strong>of</strong> card<strong>in</strong>ality π,<br />
where each Pt satisfies the tt-cha<strong>in</strong> condition. There is no loss <strong>in</strong><br />
generality <strong>in</strong> assum<strong>in</strong>g that C C Π° m(Pt\ieI). For xeC, the sets<br />
spQ(x) each have card<strong>in</strong>ality less than m and form a cha<strong>in</strong> under<br />
<strong>in</strong>clusion; therefore, by the Erdόs-Rado theorem (a pro<strong>of</strong> <strong>with</strong>out<br />
appeal to this theorem is not difficult), \{spo(x)\xe C}\
CHAIN CONDITIONS IN FREE PRODUCTS OF LATTICES 113<br />
be an m-homomorphism such that ψ(β) = 0 and ψ(β + 1) = 1. We<br />
now def<strong>in</strong>e the m-homomorphism φ: L —• B t ϋ {0,1} by φ(x) if x e B 19<br />
and φ(x) = ψ(x) if xe B 2. S<strong>in</strong>ce φ((x Vβ) Λ (β + 1)) = #, it now<br />
follows that I£7/1 = n β. Therefore, \C\ = n, complet<strong>in</strong>g the pro<strong>of</strong>.<br />
Theorem 4 is easier to prove. Indeed, if n (ajσ( i), , #„(#!, ••-,!/») be proper representations <strong>with</strong> « V%} We can assume there is an <strong>in</strong>teger k <strong>with</strong> 0 ^ k
114 G. GRATZER, A. HAJNAL AND DAVID KELLY<br />
Let F be a nontrivial variety <strong>of</strong> fc^-<strong>lattices</strong> and let L be a F<strong>free</strong><br />
lattice generated by an <strong>in</strong>f<strong>in</strong>ite set X. We show that, <strong>in</strong><br />
contrast <strong>with</strong> the m = ^ 0 case, permutations <strong>of</strong> X can create dist<strong>in</strong>ct<br />
comparable elements <strong>in</strong> L. Let p and q be ^-polynomials <strong>in</strong><br />
variables {x n \ n < ft)} such that p ^ q holds <strong>in</strong> F (for any substitution)<br />
but p = q does not (for example, x 0 and x 0 VΛΓJ. Let x\ be<br />
dist<strong>in</strong>ct elements <strong>of</strong> X for is Z (the <strong>in</strong>tegers) and n < ω. Further,<br />
let Pi = p(x\\n
CHAIN CONDITIONS IN FREE PRODUCTS OF LATTICES 115<br />
for x 6 X otherwise extends to an τn-homomorphism <strong>of</strong> L onto 2<br />
that maps y a to 0 and y β to 1; thus, y a Φ y β.<br />
Problem 3. Is every m-complete cha<strong>in</strong> conta<strong>in</strong>ed <strong>in</strong> a Boolean<br />
m-algebra <strong>in</strong> DJ<br />
If m = xt + , a Boolean m-algebra <strong>in</strong> D m is called n-representable<br />
by R. Sikorski [10]. If, for any two dist<strong>in</strong>ct elements <strong>of</strong> an mlattice<br />
L, there is an m-homomorphism from L onto 2 separat<strong>in</strong>g<br />
the two elements, then L is <strong>in</strong> D m. Thus, as observed <strong>in</strong> the pro<strong>of</strong><br />
<strong>of</strong> Lemma 3, any successor ord<strong>in</strong>al is an m-sublattice <strong>of</strong> a power<br />
set. It also follows that D m conta<strong>in</strong>s every m-complete cha<strong>in</strong>.<br />
(Replace each element <strong>of</strong> an m-complete cha<strong>in</strong> C by two elements,<br />
form<strong>in</strong>g the cha<strong>in</strong> C; then C is an m-sublattice <strong>of</strong> a power set and<br />
the obvious map from C to C is an m-homomorphism.) S<strong>in</strong>ce the<br />
embedd<strong>in</strong>g <strong>of</strong> a cha<strong>in</strong> <strong>in</strong>to the Boolean algebra that it ϋί-generates<br />
preserves all exist<strong>in</strong>g jo<strong>in</strong>s and meets (see [5]), any m-complete<br />
cha<strong>in</strong> is an m-sublattice <strong>of</strong> a Boolean m-algebra. However, the<br />
follow<strong>in</strong>g example shows that m-congruences <strong>of</strong> maximal cha<strong>in</strong>s need<br />
not extend to m-congruences <strong>of</strong> Boolean m-algebras. (Contrast <strong>with</strong><br />
the m = fc$ 0 case <strong>in</strong> [5].) Let B be the power set <strong>of</strong> [0,1] and let<br />
C be the maximal cha<strong>in</strong> <strong>in</strong> B consist<strong>in</strong>g <strong>of</strong> all <strong>in</strong>tervals <strong>of</strong> the<br />
form [0, x) or [0, x], where xe[0, 1]. The m-homomorphism that<br />
only collapses [0, x) and [0, x],0^x^l, maps C onto [0, 1], Yet,<br />
if m ^ (2**°)+, any m-congruence <strong>of</strong> B that collapses [0, x) and [0, x],<br />
0 ^ x ^ 1, collapses all <strong>of</strong> B s<strong>in</strong>ce [0, 1] C U([0, x] - [0, x)\0^x£l.)<br />
REFERENCES<br />
1. M. E. Adams and D. Kelly, <strong>Cha<strong>in</strong></strong> <strong>conditions</strong> <strong>in</strong> <strong>free</strong> <strong>products</strong> <strong>of</strong> <strong>lattices</strong>, Algebra<br />
Universalis, 7 (1977), 235-243.<br />
2. W. W. Confort and S. Negrepontis, The Theory <strong>of</strong> Ultrafilters, Spr<strong>in</strong>ger-Verlag,<br />
New York, 1974.<br />
3. P. Erdδs and R. Rado, Intersection theorems for systems <strong>of</strong> sets II, J. London<br />
Math. Soc, 44 (1969), 467-479.<br />
4. F. Galv<strong>in</strong> and B. Jόnsson, Distributive suh<strong>lattices</strong> <strong>of</strong> a <strong>free</strong> lattice, Canad. J. Math.,<br />
13 (1961), 265-272.<br />
5. G. Gratzer, General Lattice Theory, Birkhauser Verlag, Basel, 1979.<br />
6. G. Gratzer and D. Kelly, Free m-product <strong>of</strong> <strong>lattices</strong>, Colloq. Math., to appear.<br />
7. G. Gratzer and D. Kelly, A normal form theorem for <strong>lattices</strong> completely generated<br />
by a subset, Proc. Amer. Math. Soc, 67 (1977), 215-218.<br />
8. G. Gratzer and H. Lakser, <strong>Cha<strong>in</strong></strong> <strong>conditions</strong> <strong>in</strong> the distributive <strong>free</strong> product <strong>of</strong><br />
<strong>lattices</strong>, Trans. Amer. Math. Soc, 144 (1969), 301-312.<br />
9. B. Jόnsson, Relatively <strong>free</strong> <strong>products</strong> <strong>of</strong> <strong>lattices</strong>, Algebra Universalis, 1 (1972), 362-373.<br />
10. R. Sikorski, Boolean Algebras, 3rd edition, Spr<strong>in</strong>ger-Verlag, New York, 1969.<br />
Received March 21, 1978 and <strong>in</strong> revised form December 29, 1978. The research <strong>of</strong><br />
the authors was supported by the National Research Council <strong>of</strong> Canada.<br />
UNIVERSITY OF MANITOBA<br />
WINNIPEG, CANADA
DONALD BABBITT (Manag<strong>in</strong>g Editor)<br />
University <strong>of</strong> California<br />
Los Angeles, California 90024<br />
HUGO ROSSI<br />
University <strong>of</strong> Utah<br />
Salt Lake City, UT 84112<br />
C. C. MOORE and ANDREW OGG<br />
University <strong>of</strong> California<br />
Berkeley, CA 94720<br />
E. F. BECKENBACH<br />
PACIFIC JOURNAL OF MATHEMATICS<br />
UNIVERSITY OF BRITISH COLUMBIA<br />
CALIFORNIA INSTITUTE OF TECHNOLOGY<br />
UNIVERSITY OF CALIFORNIA<br />
MONTANA STATE UNIVERSITY<br />
UNIVERSITY OF NEVADA, RENO<br />
NEW MEXICO STATE UNIVERSITY<br />
OREGON STATE UNIVERSITY<br />
UNIVERSITY OF OREGON<br />
EDITORS<br />
ASSOCIATE EDITORS<br />
J. DUGUNDJI<br />
Department <strong>of</strong> Mathematics<br />
University <strong>of</strong> Southern California<br />
Los Angeles, California 90007<br />
R. FINN AND J. MILGRAM<br />
Stanford University<br />
Stanford, California 94305<br />
B. H. NEUMANN F. WOLF K. YOSHIDA<br />
SUPPORTING INSTITUTIONS<br />
UNIVERSITY OF SOUTHERN CALIFORNIA<br />
STANFORD UNIVERSITY<br />
UNIVERSITY OF HAWAII<br />
UNIVERSITY OF TOKYO<br />
UNIVERSITY OF UTAH<br />
WASHINGTON STATE UNIVERSITY<br />
UNIVERSITY OF WASHINGTON<br />
Pr<strong>in</strong>ted <strong>in</strong> Japan by International Academic Pr<strong>in</strong>t<strong>in</strong>g Co., Ltd., Tokyo, Japan
Pacific Journal <strong>of</strong> Mathematics<br />
Vol. 83, No. 1 March, 1979<br />
Richard Neal Ball, Topological lattice-ordered groups . . . . . . . . . . . . . . . . . . . . 1<br />
Stephen Berman, On the low-dimensional cohomology <strong>of</strong> some<br />
<strong>in</strong>f<strong>in</strong>ite-dimensional simple Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27<br />
R. P. Boas and Gerald Thomas Cargo, Level sets <strong>of</strong> derivatives . . . . . . . . . . . . 37<br />
James K. Deveney and John Nelson Mordeson, Splitt<strong>in</strong>g and modularly<br />
perfect fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45<br />
Robert Hugh Gilman and Ronald Mark Solomon, F<strong>in</strong>ite groups <strong>with</strong> small<br />
unbalanc<strong>in</strong>g 2-components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55<br />
George Grätzer, Andras Hajnal and David C. Kelly, <strong>Cha<strong>in</strong></strong> <strong>conditions</strong> <strong>in</strong> <strong>free</strong><br />
<strong>products</strong> <strong>of</strong> <strong>lattices</strong> <strong>with</strong> <strong>in</strong>f<strong>in</strong>itary operations. . . . . . . . . . . . . . . . . . . . . . . . 107<br />
Benjam<strong>in</strong> Rigler Halpern, Periodic po<strong>in</strong>ts on tori . . . . . . . . . . . . . . . . . . . . . . . . 117<br />
Dean G. H<strong>of</strong>fman and David Anthony Klarner, Sets <strong>of</strong> <strong>in</strong>tegers closed under<br />
aff<strong>in</strong>e operators—the f<strong>in</strong>ite basis theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 135<br />
Rudolf-Eberhard H<strong>of</strong>fmann, On the sobrification rema<strong>in</strong>der s X − X . . . . . . . 145<br />
Gerald William Johnson and David Lee Skoug, Scale-<strong>in</strong>variant<br />
measurability <strong>in</strong> Wiener space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157<br />
Michael Keisler, Integral representation for elements <strong>of</strong> the dual <strong>of</strong><br />
ba(S, ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177<br />
Wayne C. Bell and Michael Keisler, A characterization <strong>of</strong> the representable<br />
Lebesgue decomposition projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185<br />
Wadi Mahfoud, Comparison theorems for delay differential equations. . . . . . 187<br />
R. Daniel Mauld<strong>in</strong>, The set <strong>of</strong> cont<strong>in</strong>uous nowhere differentiable<br />
functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199<br />
Robert Wilmer Miller and Mark Lawrence Teply, The descend<strong>in</strong>g cha<strong>in</strong><br />
condition relative to a torsion theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207<br />
Yoshiomi Nakagami and Col<strong>in</strong> Eric Sutherland, Takesaki’s duality for<br />
regular extensions <strong>of</strong> von Neumann algebras . . . . . . . . . . . . . . . . . . . . . . . . 221<br />
William Otis Nowell, Tubular neighborhoods <strong>of</strong> Hilbert cube manifolds . . . . 231<br />
Mohan S. Putcha, Generalization <strong>of</strong> Lent<strong>in</strong>’s theory <strong>of</strong> pr<strong>in</strong>cipal solutions <strong>of</strong><br />
word equations <strong>in</strong> <strong>free</strong> semigroups to <strong>free</strong> product <strong>of</strong> copies <strong>of</strong> positive<br />
reals under addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253<br />
Amitai Regev, A primeness property for central polynomials . . . . . . . . . . . . . . 269<br />
Saburou Saitoh, The Rud<strong>in</strong> kernels on an arbitrary doma<strong>in</strong>. . . . . . . . . . . . . . . . 273<br />
He<strong>in</strong>rich Ste<strong>in</strong>le<strong>in</strong>, Some abstract generalizations <strong>of</strong> the<br />
Ljusternik-Schnirelmann-Borsuk cover<strong>in</strong>g theorem . . . . . . . . . . . . . . . . . . 285