Mathematica Pannonica13/2 (2002), 183-190

A NOTE ON THE TOLERANCE LAT-TICE OF ATOMISTIC ALGEBRAICLATTICES

Sandor RadeleczkiInstitute of Mathematics, University of Miskolc, 3515 Miskolc-Egyetemvriros, Hungary

Dedicated to Professor R. Wiegandt on his 70th birthday

Received: April 2001

MSC 2000: 06 B 10,06 D 15

Keywords: Standard element,' tolerance-trivial and pseudo complemented lat-tice.

, t

Abstract: We prove that ~ny tolerance of an algebraic atorpistic lattice L\

with a regular kernel is a congruence. Moreover the lattices TolL and ConLsatisfy the same identities involving the pseudo complementation.

1. Introduction

It is a known fact that the tolerance lattice (TolL) and the con-gruence lattice (ConL) of a finite atomistic lattice coincide. In generalthis assertion is not true in the infinite case, however we show that if Lis an atomistic algebraic lattice then any tolerance of L, whose kernelis a principal ideal, is a congruence.

It is also well-known [1] that TolL and ConL are pseudocomple-mented lattices. Let T* (or e*) stands for the pseudo complement of a

E-mail address:matradi@gold-unimiskolc.huResearch partially supported by Hungarian National foundation for Scientific Re-search (Grant No. T029525 and No. T034137) and by Janos Bolyai Grant ofHungarian Academy of Science

184 S. Radeleczki

T E Tol L (or of a e E Con L). Bounded pseudo complemented latticesconsidered as universal algebras (L, /\, V,* ,0,1) satisfying the identity

(Xl /\ X2 /\ ... /\ xn)*VV(x~ /\ X2 /\ ... /\ xn)* V ... V (Xl /\ x2 /\ ... /\ x~)* = 1

form an equational class for any n E N. Any equational class of dis-tributive bounded pseudo complemented lattices coincides with one ofthese classes (see [7]). Our principal result is the following:Main Theorem. If L is an atomistic algebraic lattice, then Con Lsatisfies the identity (Ln) for some n E N if and only if Tol L satisfiesthe same identity.

2. Preliminaries

The lattice L with 0 element is called atomistic if any X E L is thejoin of atoms below x. The set of all atoms of L is denoted by A(L)and for X E L let A(x) = {a E A(L) I a ::; x}. Clearly, A(x /\ y) == A(x) n A(y) and A(x) U A(y) ~ A(x V y) for all x, y E L. Let (x]stand for the principal ideal generated by x E L.

An element a E L is called standard if x /\ (a Vy) = (x /\ a) V (x /\ y)holds for all x, y E L. The standard elements of L form a sublatticedenoted by S(L) (see [4]).Remark 2.1.The following simple facts can be found e.g. in [4] or [5].

(i) s is a standard element of L iff the relation e(s) = {(x, y) EE L2 I x V y = (x /\ y) Va, for some a ::; s} is a congruenceof L.

(ii) For any Sl, S2 E S(L) we have e(Sl V S2) = e(Sl) Ve(S2),e(Sl/\ S2) = e(Sl) /\ e(S2) and e(Sl) 0 e(S2) = e(S2) 0 e(Sl)'

Let T(a, b) stand for the principal tolerance generated by the pair(a, b) E L2, a =1= b. For the identical and the unit relations on L, wewrite 6. and \l, respectively. If T E Tal L, then for any a, bEL we have(a, b) E T {:} (a/\b,aVb) E T and (a,b) E T, a ~ x ~ b => (a,x), (x, b) EE T (see [2]). Clearly, ConL ~ TolL and for any el, e2 E ConL theirgreatest lower bound in TolL is the same as the "meet" el/\ e2 E Con L.However this is not true for the "join" operation; denoting by Tl U T2the least upper bound ofTl, T2 E TolL in TolL, we have elUe2 ~ el Ve2for all el, e2 E Con L. If TolL = ConL, then L is called tolerance-trivial.

A note on the tolerance lattice 185

Lemma 2.2. Let L be a lattice with ° and 8 E S(L). Then(i) 8(8) = T(O, 8),(ii) For every 81,82 E S(L) we have 8(81) U 8(82) = 8(81) V 8(82)'

Proof. (i) Obviously, 8(8) E TolL and (0,8) E 8(8). Take any T EE TolL with (0,8) E T. Then we have (0, a) E T for each a ::; 8,whence we obtain 8(8) = {(x, y) E L2 I x V y = (x 1\ y) Va, for somea::; 8} ~ {(x, y) E L2 I (x 1\ y, x Vy) E T} = T. Hence 8(8) = T(O, 8).

(ii) In view of the above (i) we get 8(81) V 8(82) = 8(81 V 82) == T(O, 81 V 82). On the other hand, (0,81) E 8(81) and (0,82) E 8(82)implies (0,81 V 82) E 8(81) u 8(82)' whence we obtain T(O, 81 V 82) ~~ 8(81) U 8(82) ~ 8(81) V 8(82) = T(O, 81 V 82), and this proves (ii). 0Remark 2.3. An easy consequence of the above (ii) is that

8(81) U ... U 8(8n) = 8(81) V ... V 8(8n), for all81 , ... , 8n E S(L).Let T denote the transitive closure of aTE TolL, it is well-knownthat T E ConL and that T(O) = {x ELI (O,x) E L} is an ideal of L(see [8)). For an atomistic algebraic lattice L we shall make use of thefollowing notations: A(T) = A(L) n T(O) = {a E A(L) I (0, a) E T}and WT = V{a I a E A(T)}.Remark 2.4. It is easy to check that for any family Ti E TolL, i E I,

we have A (1\ Ti) = n A(Ti).iEI iEI

The following result will play an important role'in our develop-ment:Proposition A ([9], Prop. 1.11). For each T E TolL we have A(T) == A (WT) and WT i8 a 8tandard element of L.

A lattice L with ° is called a pseudocomplemented lattice if foreach x E L there exists an x* E L such that for any y E L, y 1\ x = ° {::>

{::> y ~ x*. If x* V x** = 1 for all x E L, then L is called a Stone lattice.A lattice L is said to be a O-modular if, for any a, b, eEL, a ~ c andb 1\ c = ° imply (a V b) 1\ c = a (as given in [10)). According to [1], TolLis a pseudo complemented O-modular lattice for any lattice L.

3. Tolerances with a regular kernel

Definition 3.1. We say that a tolerance T E TolL has a regular kernelif T(O) is a principal ideal of L.

Clearly, if L satisfies the ascending chain condition then any tol-erance of it has a regular kernel. (Indeed, in this case any ideal of L is a

186 S. Radeleczki

principal ideal [3].) It is also obvious that any congruence 8(s) inducedby a standard element s E L is a tolerance with a regular kernel.Proposition 3.2. If L is an atomistic algebraic lattice, then any toler-ance T with a regular kernel is a congruence of L induced by a standardelement of L.Proof. Let T E TolL be a tolerance with a regular kernel. As T(O) == (u] for some U E L, Prop. A gives A(WT) = A(T) = A(u) implyingu = WT. Thus (O,WT) E T, whence by Lemma 2.2(i) we get e(WT) == T (0, WT) ;£ T.

Conversely, take (x, y) E T. Then (x 1\y, x V y) E T and for eacha E A (x V y) \ A (x 1\ y) we get that (0, a) = ((x 1\ y) 1\ a, (x V y) 1\ a) E T,i.e. a E A(T). Since A(T) = A(WT), we have a ;£ WT. Hence weobtain q = V{a I a E A(x V y) \ A(x 1\ y)} ;£ WT. Now the relations(0, q) E 8(WT) and x Vy = (x 1\ y) V q imply (x Vy, x 1\ y) E 8(WT), thatis (x,y) E 8(WT)' Hence T;£ e(WT), and this proves T = 8(WT). 0

A tolerance (congruence) <pis called complete, if for any (ai, bi) EE <p,i E I, we have

(Vai,Vbi) E <p andiEI iEI

(see e.g., [11]).Corollary 3.3. (i) Any atomistic lattice L satisfying the ascendingchain condition is tolerance-trivial.

(ii) Any complete tolerance of an atomistic algebraic lattice L isa congruence induced by a standard element of L.Proof. (i) As any lattice with ° satisfying the ascending chain conditionis an algebraic lattice (see e.g., [3]), L is an atomistic algebraic lattice.Since all T E Tol L now have regular kernels, we get TolL ~ Con L, i.e.TolL = ConL.

(ii) If T E Tol L is a complete tolerance, then WT = V {a E A(L) II (0, a) E T} E T(O), whence T(O) = (WT]' Therefore T has a regularkernel, and so Prop. 3.2 gives (ii). 0

(/\ ai, /\ b'i) E <piEJ iEI

4. Pseudocomplements in the tolerance lattice

In what follows let L denote an atomistic algebraic lattice. Onthe set A(L) we define a relation R as follows: for a, bE A(L), (a, b) EE R {::} e(O, a) 1\ 8(0, b) =1= 6, For any set B ~ A(L) let R(B) = {x EE A(L) I (b, x) E R for some b E B}. In [9] we proved the following:

A note on the tolerance lattice 187

Proposition B ([9], Prop. 3.2). For any T E TolL, T* is a congruenceand we have T* = B(s) for s = V{a I a E A(T*)} E S(L) and A(T*) == A(L) \ R(A(T)).Corollary 4.1. For arbitrary TI, T2 E TolL, A(TI) = A(T2) impliesT{ = T2.Proof. A(TI) = A(T2) implies R(A(TI)) = R(A(T2)), whence in viewof Prop. B we get A(T{) = A(Tn and this gives T{ = T2· 0

For any pseudo complemented lattice I: its Boolean part is definedas B(I:) = {x E I: I x = x**}. (B(I:), I\,~) is a Boolean algebra, wherea ~ b is defined to be (a* 1\ b*)* (for more details, see [6]). Now weformulate the followingCorollary 4.2. B(ToIL) and B(ConL) are the same Boolean alge-bras.Proof. Clearly, B(ConL) ~ B(ToIL). Take any T E B(ToIL). As T == (T*)*, Prop. B gives T E ConL and T = T** implies T E B(ConL).Hence we get B(ToIL) = B(ConL) and obviously the correspondingBoolean algebras are the same. 0Proposition 4.3. Let L be an atomistic algebraic lattice. If Tol L isa complemented lattice, then L is tolerance-trivial, Con L is a Booleanlattice and any <p E ConL is a factor congruence of L.Proof. Take any T E Tol L. Denoting the complement of T by T', firstwe prove that (T')* = T. Since T I\T' = 6 implies T ;;; (T')* and sinceTol Lis O-modular, we obtain (T')* = (TUT')I\(T')* = T. Now Prop. Bgives T = B(s) E ConL for some s E S(L), i.e. we get that TolL == Con L. Since Con L is distributive and complemented, it is Boolean.Let <p E Con L, then <p has a complement <p', moreover <p = B(S I) and<p' = B(S2) for some Sl, S2 E S(L). As in view of Remark 2.1(ii) wehave B(SI) 0 B(S2) = B(S2) 0 B(SI), <p and <p' permute, therefore they arefactor congruences of L. 0

5. The proof of the main result

To prove our Main Theorem we need the followingLemma 5.1. ([9], Lemma 1.12). (i) For each T E TolL, A(T)= A(T).

(ii) For any O!i E Con L, i E I we have

188 S. Radeleczki

Corollary 5.2. For any T E TolL we have (T)* = T*.Proof. As the above (i) gives A(T) = A(T), by applying Cor. 4.1 wededuce (T)* = T*. 0Proof of Main Theorem. First, assume that Tol L satisfies the iden-tity (Ln) and consider e1,e2, ... ,en E ConL. As (ConL,/\,*) is asub algebra of (Tol L, /\, * ), we can write(611/\ 612/\",/\ en)* V (e~ /\ 612/\",/\ en)* V V (611/\ 612/\",/\ e~)* ~

(611/\612/\" ./\en)*u(e~ /\612/\" ./\en)*u u(e1/\e2/\ ... /\e~)* = \7,and this proves that ConL satisfies the identity (Ln).

Conversely, assume that ConL satisfies (Ln) and consider T1,

T2"'" Tn E TolL. As Ti E ConL, 1 ~ i ~n, we have:(Tl /\ T2 /\ ... /\ T n)*V

V((Tl)* /\ T2 /\ ... /\ T n)* V ... V (Tl /\ T2 /\ ... /\ (T n)*)* = \7.

Let us introduce the notations: /30 = Tl /\ T2 /\ ... /\ Tn, /31 == (Tl)* /\T2/\· .. /\Tn, /3n = T1/\T2/\·· ./\(Tn)*. Since by Cor. 5.2 wehave T{ = (Tl)*' T2 = (T2)*" .. , T~ = (T n)*, in view of Remark 2.4and applying Lemma 5.1 we obtain

n

i=1n

= nA(Td = A(TI /\ T2 /\ ... /\ Tn) = A(/3o),i=1

A(T{ /\ T2 /\ /\ Tn) = A(Tn n A(T2) n ... n A(Tn) ==A((Tl)*) n A(T2) n n A(T n) =A((T1)* /\ T2 /\ ... /\ Tn) =A(/31),

A(TI /\ T2 /\ ... /\ T~) = A(TI/\ T2 /\ ... /\ (T n)*) = A(/3n).Now, Cor. 4.1 gives that (T1/\T2/\ ... /\Tn)* = /3~, (T{ /\T2/\···/\

/\ Tn)* = /31, ... ,(Tl /\ T2 /\ ... /\ T~)* = /3~. Since in view of Prop. Bthere exist So, SI, "Sn E S(L) such that /3~= e(so), f3i = e(Sl)' , /3~== e(sn), by Remark 2.3 we have /3~U /3i u ... u /3~= f3~V /3i V V /3~.Summarizing the above results we can write:

A note on the tolerance lattice 189

(Tl /\ T2 /\ /\ Tn)*U

U(T{ /\ T2 /\ ... /\ Tn)* U U (Tl /\ T2 /\ ... /\ T~)* =

= f3~ U f3~ U ... U f3~ = f3~ V f3~ V ... V f3~ =

= (Tl/\ T2/\···/\ Tn)*V

V((Tl)* /\ T2 /\ ... /\ T n)* V ... V (Tl /\ T2 /\ ... /\ (T n)*)* = V,thus TolL satisfies the identity (Ln). 0Corollary 5.3. Let L be an atomistic algebraic lattice. Then Tol L isa (not necessarily distributive) Stone lattice if and only if ConL is aStone lattice.Proof. Substituting n = 1 in (Ln) we obtain the Stonean identityx* V x** = 1. 0Corollary 5.4. If L is a weakly modular atomistic algebraic lattice,then Tol L is a (not necessarily distributive) Stone lattice.Proof. Since the congruence lattice of a weakly modular atomistic al-gebraic lattice is a Stone lattice (as noted by [9]), our assertion is animmediate consequence of Cor. 5.3. 0

Problems. Let L denote an atomistic algebraic lattice.1) Under what conditions TolL is distributive?2) Under what conditions we have Con(ToIL) rvCon(ConL) ?

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190 S. Radeleczki: A note on the tolerance lattice

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[11] WILLE, R.: Complete tolerance relations of concept lattices, Preprint No. 794Techn. Hochschule Darmstadt (1983) 1-23.