Algebra Universalis, 7 (1977) 163-169 Birkh~iuser Verlag, Basel

Splitting lattices generate all lattices

by

ALAN DAY*

w Introduction

In [2], McKenzie introduced the concepts of a bounded lattice homomorph i sm and of a splitting lattice. In that paper, he posed a problem (#6) as to whether or not the variety of all lattices was generated by the class of all splitting lattices. An equivalent formulation is whether or not every proper quotient in a finitely generated free lattice contains a prime quotient. The purpose of this paper is to provide an affirmative answer to that problem.

Our approach to the question is somewhat indirect. For a given lattice we construct an inverse limit which produces a lattice satisfying Whi tman ' s fourth condition (see [1], [2] or [3]). We then note that, if the original lattice were a finite bounded homomorphic image of a free lattice, each member of the inverse limit system is again such. Finally, by starting with a suitable such lattice, we show that the inverse limit contains a copy of the FL(3), the free lattice on three generators.

w Preliminaries

Let us first recall some definitions and results from [2]. A lattice homomorph- ism qS:A ~ B is called upper (resp. lower) bounded if, for all b ~ B, 4~-1[b] # 4~ implies it has a greatest (resp. least member) , q5 is bounded if it is both upper and lower bounded. We let ~ be the class of all finite lattices that are bounded homomorphic images of finitely generated free lattices.

One notes that tb :A , , B is bounded, if and only if 4~ has left and right adjoints o~ : B ~ A and /3 : B ~ A respectively satisfying

b<-c~(a) iff a(b)<-a

d~(a)<-b iff a~lS(b)

* This research was supported by NRC grant A8190. Presented by G. Gfiitzer. Received November 26, 1975. Accepted for publication in final form May 10, 1976.

163

164 ALAN DAY ALGEBRA UNIV.

It is clear that both a and /3 are injective with a being join preserving and /3 being meet-preserving. The main result that we need from [2] is:

T H E O R E M . Let L be a lattice. If there is one homomorphism from a finitely generated free lattice onto L that is upper bounded (resp. lower bounded or bounded) then every homomorphism into L from a finitely generated lattice is again such.

This result allows us to prove a lattice, L, belongs to ~ by exhibiting one upper bounded and one lower bounded epimorphism from a finitely generated free lattice onto L.

We also need a lattice constructio.n found in Day [1]. Let L be a lattice and I=[u ,v] an interval in L. We construct a new lattice by letting L[I]--- (LkI)t.J ( Ix2 ) and defining x - y in L[I] iff one of the following hold

(1) x , y ~ L \ I and x - y i n L

(2) x = (a, i), y e L \ I and a-< y in L

(3) x ~ L \ I , y = ( b , j ) and x < - b i n L

(4) x = ( a , i ) , y = ( b , j ) and a - < b i n L and i - - - j in2

There is a natural epimorphism Kr:L[I] ~ L with

K~(x) = { x x ~ L \ I x = ( a , i )

Moreover K~ is bounded below and above by a :L> -~, L[I] and/3 :L ~ L[I] where

[(x, 0), x e I Or(X) /

tx, x~ I and /3(x)- [(x, 1 ) , x e I

- ~ x, x r I

w Whitman covers

Whitman's fourth condition in his solution of the word problem for free lattices is the following universal sentence:

(W) ( a ^ b < _ c v d ) ~ { a , b , c , d } N [ a ^ b , c v d ] ~ f b

Therefore in any lattice L we can define

W[L] ={(a, b, c, d ) e L4:(a, b, c, d) I~ (W)}

Vol 7, 1977 Splitting lattices generate all lattices 165

Now the construction from [1] used an interval I=[aAb , c vd] where (a,b,c ,d)~ W[L] to create the lattice L[I] in which a A b ~ - c v d in L[I]. Moreover if (p ,q ,r , s )eW[L] with p ^ q = a A b and r v s = c v d then [pAq, r v s ] = I and in L[I], p A q ~ - r v s as well. We define

Iw(L) = {[a ^ b, c v d]: (a, b, c, d) ~ W[L]}

(3.1) LEMMA. For every lattice L not satisfying (W) there exists a lattice/2 and bounded epimorphism o : L----L satisfying: For all (al, a2, a3, a4) ~ W[ L ] and any xiep-l[ai] i = 1 , 2 , 3 , 4 , X1AX2~-X3VX 4 in /2.

Proof. For each I~.r we have KI : L [ I ] ~ - L bounded by a , :L , -~ L[I] and 131 : L~---~ L[I]. Define:

/2= N{Eq(K, -m, Kj �9 ~s):I,.l~c~w(L)} (t)

p : s L, p=(,,,. ~,)1s (2)

where zrl is the projection from the product of all the L[ I ] ' s onto the I th factor.

Now for every x e L, we have/3(x) = (/31(x)) and a(x) = (al(x)) belonging to/2. It is clear that 13 and a provide the upper and ' lower bounds respectively for p.

Now if (a, b, c, d)e W[L] then we need only show that a(a)Aa(b)~-[3(c)v /3(d) to establish our last claim. However , (a, b, c, d)~ W(L) implies [aAb, c v d] = I ~ ,r and therefore at(a) A at(b) = (a A b, 1) ~ (c v d, O) = ]3x(c) v/3x(d). This establishes the result.

Now for a lattice L define

Lo = L

Ln+l =/2,, o..5~ L~

and let L~ = lim (Ln, p,) that is: for x ~ I-[~ "~ L,,

x~L~iffVneNo pn+l(Xn+l)=Xn

Now take p,q,r, seL= satisfying the relations p $ r v s , q~-rvs, lYAq~-r and pAq~-s.

166 ALAr~ DAY AI.OEBRA umv.

Therefore there exists k, 1, m, n e No with

Pk A qk ~- rk

pz ^ qz $ st

p,,, ~- r., v s,,,

and

q,,$r, vs,,

Since each p~ : L~ --~ L~-1 is order preserving we have for any j ~ max {k, l, m, n}

pjAqjSrj

p~Aq~$s~

pj S ri v sj

and

qj$-rjvsj

Now if pjAqiSr~vsi then p A q $ r v s and we are done. If p~/xq~<-rjvs i then by the previous lemrna, p~+~/\qj+l~:-rj+lvsj+~ and again pAq~-rVS.

Therefore L= satisfies (W). Moreover since each of the p, is bounded, it follows easily that the sequences

/3(x) = (/3,(x)) and a(x )=(a , ( x ) )

provide upper and lower bounds for the epimorphism

p=o : L~--~ L = Lo.

We have therefore proven the following:

(3.2) T H E O R E M . For any lattice, L, there is a lattice IS, satisfying (W) and a bounded epimorphism tp : I~ ~ L.

One might note at this time that there are categorical formulations that make this /~ ~-~ L universal. They do not however seem exceedingly important.

Vol 7, 1977 Splitting lattices generate all lattices

w The generation of

167

(4.1) T H E O R E M . If A ~ ~ and I = [u, v] c_ A then A[I] ~ N3.

Proof. Let X be a finite set with f:FL(X)-~-~ A[I] a lattice epimorphism. If A ~ ~ then K- f : FL(X) ---~A is bounded below by a t: A ~ FL(X) and above by [3:A ~-~ FL(X). By the definition of K we have

{at, a e A\X

K-' [a ] = {(a, 0), (a, 1)} a ~ I

and therefore aK(t) provides a lower bound in FL(X) of f - l [ t ] for every t~ (A\[)(.J Ix{O}. We need only show then that f-I[(u, 1)] has a lower bound.

Since f is surjective there is a w c FL(X) with f (w)= (u,1). Now define

~ = w / x / k { x 6 X : ( u , 1)-<f(x)}/x A { a ( a ) : a 6 A \ I and a > u }

We claim that ~ is the least member of f-a[(u, 1)]. Clearly f ( ~ ) = (u, 1) and if

S = {p ~ FL(X) : (u, 1) - f(p) implies v~ _< p}

then X c S and S is closed under meets. By the construction of A[I], (u, 1 ) - < f ( pv q)= f(p)vf(q) implies one of three possibilities:

(a) (u, 1 ) - f (p)

(b) (u, 1)-<f(q)

(c) (u, 1 )<f (pvq) and f ( p v q ) ~ A \ I

The first two cases are covered by induction and the last case implies Kf(p v q)= f (pvq) and then # - - - a K f ( p v q ) < _ p v q . Therefore f is lower bounded by c~ :A[I ] ~ FL(X) where:

aK(s), s e ( A \ I ) U ( I x { O } ) ~r(s) = ~ yaK(s) , s ~ I x { 1 }

The proof that f : FL(X) ~-~ A[I] is upper bounded follows similarly.

168 ALAN DAY ALGEBRA UNIV.

(4.2) C O R O L L A R Y 1. If A e ~ then fi, e ~ where A is the lattice constructed in (3.1) from A.

Proof. ~ is closed under finite subdirect products and by (4.1) A [ I ] e ~ for each I e &~(A).

(4.3) THEOREM. .~ = I-ISP(~).

Proof. Let Bo = FD(3)e ~, the free distributive lattice on 3 generators and in B| take :7, Y, s whose projections on to Bo are the three generators, x, y and z respectively. Since {x, y, z} is join-prime and meet-prime set of generators of Bo, {x, Y, z} is a join-prime and meet-prime set of generators of (~, Y, z)<-Boo and therefore (~, Y, z)~-FL(3) follows from [3].

(4.4) C O R O L L A R Y 1. In a finitely generated free lattice, every proper quotient contains a prime quotient.

(4.5) C O R O L L A R Y 2. For any proper subvariety Y{ of ~ , there is a splitting equation e such that YC c ~ ( e ).

These last two statements were proven equivalent to the theorem statement by A. Kostinsky (see [2]).

(4.6) C O R O L L A R Y 3. Let 9O be the class of all splitting lattices. (That is 9' = {S e ~ : S is subdirectly irreducible}) Then 9o c_ HS{Bn : n e No}.

Pro@ By [2], every S e 9 o generates a completely-join-prime variety. (That is: S e V (Y~i : i e I) implies S e Y~ for some i e L) Since V {HSP(Bn) : n e No} = ~ we must have for every S e 9O, there is some n e No with S e HSP(Bn). By J6nsson's Lemma, it follows that S e I-IS(B,).

w Concluding remarks

By the results of McKenzie, the structure of members of ~ is closely related to the structure of finite sublattices of free lattices. It would be of interest then if Theorem (4.3) and in particular its corollaries could shed any light on their structure.

Vol 7, 1977 Splitting lattices generate all lattices 169

REFERENCES

[1] A. DAY, A simple solution to the word problem for lattices, Can. Math. Bull. 13 (1970), 253-254. [2] R. McKENzm, Equational bases and non-modular lattice varieties, Trans. Amer. Math. Soc. 174

(1972), 1-43. [3] B. JONSSON, Relatively free lattices, Coll. Math. 21 (1970), 191-196.

Lakehead University Thunder Bay, Ontario Canada