New Notes on tolerance relations oflatticesgratzer/images/math... · 2010. 4. 28. · A tolerance relation 6 on a lattice L is a reflexive and symmetric binary rela tion satisfying

Acta Sci. Math., 54 (1990), 229-240

Notes on tolerance relations of lattices

G. GRATZER and G. H. WENZEL *)

o. Introduction

A tolerance relation 6 on a lattice L is a reflexive and symmetric binary rela-tion satisfying the substitution property. In 1982, G. CZEDU [1] proved that, for alattice L and a tolerance relation 6, the maximal 6-connected subsets of L forma lattice. He considered lattices as algebras of type (2, 2) and gave an algebraicproof. In Section 1, we investigate tolerances from the point of view of partialordering in detail; in particular, we give an order-theoretical proof of Czedli's result.Our proof avoids Zorn's axiom needed by Czedli. Some results on 6-block fixingsets and consequences thereof are added.

Tolerances can be viewed as quotients of congruences in a natural way. Usingthis fact, we extend the Second Isomorphism Theorem from congruences to tolerancerelations in Section 2. In connection with the extended Second Isomorphism Theo-rem, a question on the product of lattice varieties arises naturally_In Section 3we answer it partially and illustrate the situation with examples.

1. The lattice LI6

For concepts and notations not defined in this paper, see G. GRATZER [3].Let L=(L; be a lattice and 6 a tolerance relation on L. x6y (x, yEL)denotes, as usual, that (x,y)E6 holds; also, H1eHg (H1 , Hg~L) denotes thatxey holds for every xEH1 , yEH,,- The following two lemmata are useful inmany situations.

Lemma 1. Let and y;§y' be elements of L with x'ex, y'ey, xey',yex'. Then (x'Vy')6(xAy).

*) The research of both authors was supported by the NSFRC of Canada.Received September 5, 1987.

1

230 G. Gratzer and G. H. Wenzel

that holds.

Lemma 2. Let x, y, x', l be elements of L with x€Jx', yey' and x,. Then xey and x'€Jy'.

z1vtjQ

:r!./ .11'tPD c DO a01 a a a 0~.·1. ... D DOOC

zv~y

flit1fJrrnm 2

Notes on tolerance relations of lattices 231

Proof. x9x, y9y' imply that (xVy)9(xVy'). From xVy'§x, y'=f=xVy',we conclude that x9y'. The other assertion follows analogously.

We use the following notations and terminology.. A subset H of L is called9-eonnected if x9y holds for all x, yEH. If H is an arbitrary subset of L, thenwe define CH:={xEL; x9h for all hEH}. CH is either empty or it is a convexsublattice of L. CHis not necessarily 9-connected. We further define

(H]:= {xEL; x ~ h for some hEH} and [H):= {xEL; x ~ h for some hEH}.

Finally, H8:=CHn(H] and H8:=CHn[H).

Lemma 3. Let H be a subset of the lattice L.(1) H8 is a 9-connected, convex, A-closed subset ofL. IfH is upward directed,

then H8 is either empty or it is a sublattice ofL.(2) H8 is a 9-connected, convex, V-closed subset ofL. IfH is downward directed,

then H8 is either empty or it is a sublattice of L.

Proof. We only prove (1). H8 is clearly convex and A-closed. To show thatit is 8-connected, let x, yEH8 . There are x', y'EH with x~x', y=f=y', x9x',y9x', x8y', y8y'. By Lemma 1, x9y. If H is upward directed, we can choosex=y' and obtain xVy=f=X', hence xVYE(H]. Since xVyECH is clear, we getxVyEH8 ·

Lemma 4. Let H be a 9-connected subset of L.(1) (H8)8=H8 and (H8)8=H8 ·(2) H~H8~'


and A;§B are

we get ul\xE«H9)9)e. From u8«He)9)9, we get uE «(H9)9)S)9= (H8t . Hence,(H6)9 is a maximal e-connected subset of L.

In view of the last lemma, we call subsets of the form (H8)8 for 8-connectedsubsets H of L B-blocks ofL. The B-blocks are convex sublattices of L. They enjoya useful property with respect to two natural preorderings on L. In order to proveit, we use the next trivial lemma.

Lemma 6. For A, define

AVB:=~Vb; aEA, bEB}and

AAB := {aAb; aEA, bEB}.

IfA and Bare e-connected, then so are AVBand AAB.

Definition 1. For A, B?i:.L we define the following three binary relations:(1) Ao;§B:~ For all bEB there is an aEA with a;§b.(2) A ;§oB:~ For all aEA there is a bEB with a;§b.(3) A;§B:¢> Ao~B and A:§:oB.

In general, the relations =:§o and o~ are distinct. On convex subsets of L,the relation =:§ is a partial ordering. For e-blocks, the three relations coincide:

Lemma 7. If A, Bare e-blocks of L, then Ao;§B, Aequivalent.

Proof. Assume that A;§oB, i.e., for every aEA, there is a bEB with a;§b.Hence, a=aAbEAAB. Thus, A ?i:.AAB. Since AAB is 8-connected by Lemma 6and A is a maximal e-connected subset of L, we conclude that A=AAB. Hence,if bEB is given, then bAaEA for all aEA. Thus, Ao~B. The converse is anal-ogous.

Theorem 1 (see G. CZEDU [1]). If 8 is a tolerance on the lattice L, then LIe,the set of B-blocks, forms a lattice with respect to the ordering ;§. In addition. wehave AVB=(AVB)e and AAB=(AAB)9' for all A, BELie.

Proof. We prove that AVB exists and equals (AVB)9; the second formulafollows by duality. If C~A,B for a 8-block C, then trivially C~oAVB, henceC~o(AVB)9. Assuming that (AVB)9 has been shown to be a 8-block, we arefinished, since then C?!!£(AVB)fJ and, hence, (AVB)9=AV B.

Since (AVB)9 is 8-connected, we only have to show that (AVB)9 is a maxi-mal 8-connected set. Let D~(AVB)9 be a8-connected subset of L. By Lemma 6,DAA and DAB are B-connected sets. Since AVB ~D, we obtain A ~DAAand B~DAB and, hence, A=DAA, B=DAB. For dED, aEA, bEB, we get


dAaEA, dAbEB and d?!:(dAa)V(dAb)EAVB. Now d8D implies that d8(AVB),hence dE (AVB)9. Thus, D=(AVB)9, as claimed.

The description of AVBand AAB in Theorem 1 can be generalized.

Remark to Theorem 1. If AlJ A2 , ••• , A" are 8-blocks, then

A1 VA2V... VVA" = (A1VA2V ...VA,,)9and

Proof. By induction on n. For n= 1,2 we know the result. For n?!:3 we obtain

AlV... VA,,_lVA" = ((AI V... VA,,_1)VAlI)8 =

((A1V ...VAn _ 1)9VAn)9 (by the induction hypothesis) ~ (A1V...VA,,_IVA,,)8.

The maximality of AlV... VAn implies then that AlV... VA,,=(AIV...VAlI)9. Thesecond assertion follows by duality.

We add a few observations. If H~L is a non-empty, 8-connected subsetof L, then both (H9)9 and (H

9)9 are 8-blocks containing H. As the next lemmashows, the first block is the smallest and the second block is the largest 8-blockcontaining H.

Lemma 8. Let H~L be a non-empty, 8-connected set.(1) If D~H isa 8-connectedsubset of L, then H9VD~H9 and H9AD~H9.(2) If D is a 8-block with D~H, then (H9)9-;§.D-;§.(H9)9'

Proof. (1) Let xEH9V D, i.e., x=zVd for some zEH9 and dED. Thenwe have x?!:z?!:y for some yEH and x8H (since z8H and d8H hold true).Thus, XEH8, and so H9yD~H8. The proof of H8AD~H8 is analogous.

(2) D ~DA(H8VD)~DAH8~DA(H9)8 implies that D=DA(H8)8, i.e.,D-;§.(H9)8' D~DV(H9AD)~DVH8~DV(H8)8 implies D=DV(H8)8, i.e.,D?!:(H9)8.

Definition 2. IfLis a lattice and 8 is a tolerance onL, then we call a 8-con-nected subset H of L a 8-block fixing set if there exists exactly one 8-block D withH~D.

Examples. 1. If AI' ... , An are 8-blocks, then A1V...VA" and A1A...AA"are 8-block fixing sets.

Proof. Let D~AIV...VA" bea 8-block, then D?!:AIV ...VA,,=(A1V VA,,)8holds. If dED, then d8(A IV ...VA,,) and there are atEAt with d?:.thV Va,,;thus,dE(A1V...VAn)9. We obtain D~(AIV...VA,,)8 and therefore D=A1V VA".A dual argument shows that AlA...AA" is a 8-block fixing set.


2. If A, B, Care e-blocks, then (AVB)AC is not, in general, a e-blockfixing set. The following example illustrates the claim: The 8-element-Iattice L ofdiagram 3 has a tolerance e which is given by the five e-blocks A, B, C, D, E.Obviously, (AVB)AC= {y}, but {y} is not a e-block fixing set.

cB

A

Diagram 3

3. If H~L is a e-connected set, then H@ and H@ are e-block fixing sets.

Proof. If D~H@ is a e-block, then (H@)@=((H@)@)@~D~((He)e)@==(H@)@, by Lemma 8; thus, D=(H@)@, and H@ is a e-block fixing set. Dually,H@ is a e-block fixing set as well.

Theorem 2. If H~L is a e-connected set, then the following two statementsare equivalent:

(1) H is a e-block fixing set.(2) (H@)@=(H@)@.

Proof. (1) clearly implies (2), and Lemma 8 shows that (2) implies (1).

For H~L, let [H] denote the sublattice of L generated by H.

Lemma 9. Let e be a tolerance on the lattice L.(1) X, Y~L and X~oY~[X] imply X@~Y@.(2) If X, Yare e-connected subsets of L with X@ ~Y@, then (X@)@=(Ye)@.

Proof. (1) Choose aEX@. Then aex holds and there is some xEX witha~x. We choose yE Y with X-::2Y, hence a-::2y. aex implies ae[X]; thus,we have aey. We conclude aEYe .

(2) X@ and Y@ are e-block fixing sets (see Example 3 preceeding Theorem 2).Because of X@~(X@)@ and X@~(y@)e we conclude (X@)@=(Ye)@.

Lemma 9 is the crux of the next theorem.

Theorem 3. Let L be a lattice and X= {Xl' ... , xn}~L, nEN, a e-connected


finite subset of L. Then (Xetl=({xlV ...Vxn}e)e. (In words: All finitely generated8~blocks are principal 8~blocks.)

Proof. By Lemma 9,thus, (Xe)e=({XlV...Vx,,}e)8.

{XlV... Vxn} ~[XJ implies Xg ~ {XlV... VX,,}e;

2. Quotients of tolerances and the Second Isomorphism Theorem

Let L be a lattice, Con L the lattice of congruences on Land Tol L the latticeof tolerances on L. If 8 and iP are tolerances onL, then we define the binary rela~tion 8/iP on L/iP as follows: A8/iPB holds if and only if there are aEA, bEBwith a8b. If 8 and iP are congruences and 8~iP, then 8/iP is a congruence onL/iP, and the well~known Second Isomorphism Theorem states that L182;;2;;(L/iP)/{8/iP) holds. In general, 8/iP is only tolerance on L/iP. We will showthat every tolerance on an arbitrary lattice L' is of the form 8/iP for congruences8 and iP on a suitable lattice L.

Thus, let L' be a lattice and let 8' be a tolerance relation on L'. We define thelattice L as sublattice of the direct product L'X(L'/8') on the carrier setL:={(a, A); AEL'/8' and aEA}. If Itl: L-L' and 1t2: L-L'/8' are the restric~tions of the two canonical projections from L'X(I:/8') onto L' and U/8', resp.,then Itt and 1t2 are lattice epimorphisms. We define 8 :=kernel (lt2) andiP:=kemel (Itt). The homomorphism theorem yields L/iP2;;L', and the corre-sponding isomorphism identifies aEL' with 1t1l (a)EL/iP. Under this identificationwe get that L'=L/iP and 8'=8/iP.

Definition 3. Let L' be a lattice and 8' a tolerance on L'. The lattice Landthe congruences 8, iP just constructed are called the lattice, resp. the congruencesassociated with (L', 8').

We summarize:

Theorem 4. Let L' he a lattice, 8' a tolerance on L'. Let L he the lattice and8, iP the congruences associated with (L',8'). The canonical identification makesthe following two statements true:(i) L/iP = L', (ii) 8/iP = 8'.

In case of a congruence 8', we get iP=w, L=L', and 8=8' in Theorem 4.In case of a tolerance 8' we have no natural correspondence between LI8 and(L/iP)/{8IiP), but a suitable modification yields a generalized.version of the SecondIsomorphism Theorem. In order to derive the result, we find another way of inter-


preting a tolerance 6)' on a lattice L'. This interpretation associates e' on L' withiPo 80 iP on L.

Lemma 10. Let L be a lattice and 8, iP tolerances on L. Then the followingare true:

iP0804>EToIL, 8 ~ iP080iP, and iPo 80 l/>/l/> = 8jiPETolLjiP.

Quite different from situation for congruences, not every tolerance8 on a L is of the form 4>0 So iP for suitable tolerances iP, S. thetolerance 6) on the lattice L of Diagram 3 is not of that kind.

Theorem 5 (The Second Isomorphism Theorem). Let L be a lattice, iPECon L,and 8ETolL. Then

L/l/>o6)ocP ~ (LfiP)/(81cP) (LIl/»/(iP08ocPjcP).

Proof. Define L':=LlrfJ, 8':=8/rfJ, and let n: L-L' be the natural projec-tion. Ifwe extend n to the respective powersets in the canonical way, then we obtainthe mapping n: Pot(L)-Pot(L1) and, by restriction, n: L/rfJoeorfJ-pot(L1).

(i) Claim. AEL/rfJoeo cP implies that n(A)EL'/el.

Some e..block

[tI)t

[b)t

Diagram 4

AS~~llnIC that aiPo eo rfJb, acPx8ycPb for suitable xE [alrfJ, yE [blcP (a,By the definition of 8', we get ([a] rfJ)e'([b] iP). Thus, n(A) is 8'-connected. Toshow the maximality of neAl with respect to 8'-connectedness, let [x] iPEL' be anelement with ([x] iP)6)'([a] iP) for all aEA. Thus, for every aEA, there are ele-ments, x/E[x] iP, a'E[a] iP with x1eal.We conclude that xiPxl6)al cPa,xiPoeo cPa, and, hence, xiPoeo iPA. We deduce that xEA and, hence, [x] cPEn(A).Thus, n(A) is a 8'-block.

(ii) Claim. AEL'/fY implies that n-1(A)EL/po eo P.Let Clearly, n{A)=A. Choose a, bEA arbitrarily. Then

[a] iP, [b] cPEA implies that ([a] iP)61([bJ


xE[a] tP, yE[b] tP, i.e., we have atPo 80 tPb. Thus, A is tPo 80 tP-connected. Weembed A in some tP080tP-block A* and obtain n(A)=AfEn(A*). Since A and,by (i), n(A*) are 8'-blocks, we obtain A=n(A*) and, hence, A*fEn-1(A)=A.Thus, A=A*.

(i), (ii) and the surjectivity of 1t immediately imply (iii) and (iv) below.(iii) n-1(n(A»)=A holds for all tPo 80 tP-blocks A of L.(iv) n(n-1(A»)=A holds for all 8'-blocks A of L'.(v) Statements (i) to (iv) prove that the restriction ii: L/tP080tP-L'/8' is

a bijection. Finally, we show that 1i is even a lattice-homomorphism: n-1(A)== {aEL; [a] tPEA} holds for every AEL'j8'. If, therefore, A, BEL'/8' are arbi-trarily chosen, then we get:

n-1(AVB) = {xEL; [x] tPEAVB} ~~{aVb; a, bEL and [a]tPEA, [b]tPEB} = ii-1(A)Vit-1(B).

The last set is a tPo eo tP-block fixing set. Thus, there is exactly one tPo 80 tP-block of L containing it-I(A)Vn-I(B), namely n-I(A)V ii-I(B). Thus, we provedthat n-I(AVB)=it-I(A)Vit-1(B) holds. Similarly, ii-1(AAB)=ii-1(A)Ait-1(B)holds.

3. Products of lattice varieties

If V and Ware two varieties of lattices, then the product Vo W consists ofall lattices L for which there is some congruence 8 satisfying the following twoproperties:

(i) All 8-blocks of L are in V,(ii) L/8EW.

We combine these two conditions by saying that 8 establishes that L is in Vo W.G. GRATZER and D. KELLY [6] give an overview of these variety products.

VO W is not, in general, a variety. However, one knows that the variety generatedby Vo W is H(VoW), the class of all holomorphic images of lattices in VoW.R. N. McKenzie conjectured that the variety H(VoW) can be characterized asfollows: "A lattice L' is contained in H (VoW) if and only if there is a tolerance8' on L' such that all 8'-blocks of L' are in V and L'j@'EW." The next theoremanswers one direction of the conjecture in the affirmative.

Theorem 6. Let V and W be varieties of lattices. Let L' be a lattice with atolerance @' that satisfies the following two properties:

(i) All 8'-blocks are in V,(ii) L'/@'EW.

Then L'EH(VoW).


Proof. Let L be the lattice and let e, iP be the congruences associated with(L', EY). Of course, 8niP=w and L/8-=:=L'/8'. Thus, L/8EW. If 11:1: L-L'and n2 : L-L'/B' are the projections yielding iP and e, then the 8-blocks of Lare of the form n;l(Ao} for fixed AoEL'/B'. 11:;1 (Ao)= {(a, Ao); aEAo}-=:=Ao showsthat n;1(Ao)EV holds. Thus, e establishes that LEVoW. The projection n1 yieldsL'EH(L) ~H(VoW).

In order to tackle the opposite direction of the above conjecture, we beginwith some fixed lattice L'EH(VoW). L'EH(VoW) means L'===L/tP for someLEVoW and a suitable congruence if> on L. LEVoW is established by some con-gruence B on L. Then B':=B/if> is a tolerance on L', and McKenzie's conjectureseems to be based on the hope that (i) all 8'·blocks of L' are in V and (ii) L'jB'EWis always true. The next and last theorem states that the first assertion is valid.An example will show that the second one is, in general, not true. This suggeststhat the answer to McKenzie's conjecture is in the negative.

Theorem 7. Let V, W be lattice varieties and assume that L'EH(VoW).Then L'-=:=LjtP for some LEVo W and some congruence iP on L. Let 8 be a con-gruence on L establishing LEVoW. If B':=B/tP., then all 8 '-blocks ofL' are in V.

Proof. We win show that for every finite B'-connected set {["oj 4>, ... , [an] tP}of tP-blocks there is a 8-connected subset {uo' ... , un} of L with uiE[ail if>. Thissuffices, since then every 8'-block satisfies every identity which is satisfied by every8-block. Since tolerance blocks are sublattices, we may assume that [aol if>holds for all i= 1, 2, ... , n. By the definition of 8 ' , we find suitable biE[aiJ tP anda~E[ao] eJ> with a~.Due to bl~a~ and a~ea~, we get (b iVa)8a and biVaE[ai ] eJ> for i= 1,2, ... , n.With uo:=a and ui:=biVa, i=l, 2, ... ,n, our claim has been proved.

Example. We modify an example of G. CZEDU [1J to show that, under thehypotheses of Theorem 7, we cannot, in general, conclude that L'/e'EW. To doso, we describe a distributive lattice L and two congruences 8, t/J on L (with8 n t/J=w) such that Ljif>o 80 t/J (-=:=(L/t/J)!(8jfP), by Theorem 5) is not distri-butive. Let L5 be the 5-element lattice on {I, 2, 3, 4, 5} with 1


Then f/J1X f/J2 , 8 1X8 2ECon (LsXLs), and we define f/J:= f/J1X f/J2IL' 8 :=81X82IL.Diagram 5 shows the 8-blocks (indicated by bold border lines) and the f/J-blocks(indicated by normal border lines) on L.

(1,5)

Diagram 5

Diagram 6 shows the cPo 80 cP-blocks on L. We recognize that LIf/Jo 80 f/J~===N5~D.

240 G. Gratzer and G. H. Wenzel: Notes on tolerance relations of lattices

Diagram 6

Note. The conjecture referred to in this paper has in the meanwhile been an-swered in the negative by E. FRIED and G. GRATZER [2].

Note added in proof (September 19, 1990) by G. Gratzer. The second author,my former student, friend, and colleague, tragically died last year. I shall miss him.

References

[11 G. CZEDLI, Factor lattices by tolerances, Acta Sci. Math., 44 (1982), 35-42.{2] E. FRIED and G. GRATZER, Notes on tolerance relations of lattices: On a conjectureofR. McKen~

zie, Acta Sci. Math .• to appear.[3] G. GRATZER, Universal Algebra, Springer~Verlag (Berlin-Heidelberg-New York, 1979).[4J G. GRATZER, General Lattice Theory. Birkhauser Verlag (Basel, 1978).[5] G. GRATZER. The Amalgamation Property in lattice tbeory, C. R. Matll. Rep. Acad. Sci.

Canada. 9 (1987), 273-289.[6} G. GRATZER and D. KELLY, Products of lattice varieties, Algebra Universalis. 21 (1985), 33-45.[7} G. GRATZER and D. KELLY, A survey of products oflattice varieties, in: Contributions to Lattice

Theory (Proc. ConI. Szeged, Hungary, 1980), Colloquia Mathematica Societatis JanosBolyai, 33,. North-Holland (Amsterdam, 1983); pp. 457-472.

(G. G.)DEPARTMENT OF MATHEMATICSTHE UNIVERSITY OF MANITOBAWINNIPEG, MANITOBA R3T 2N2, CANADA

(G. H. W.)FAKULTAT FOR MATHEMATIK UND INFORMATIKUNIVERSITAT MANNHEIMPOSTFACH 1034626800 MANNHBIM I,BRD

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New Notes on tolerance relations oflatticesgratzer/images/math... · 2010. 4. 28. · A tolerance relation 6 on a lattice L is a reflexive and symmetric binary rela tion satisfying