Uni\JHRsitutis-ZcientioRurn

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1 Separatllm. -f~nn~l &)- Uni\JHRsitutis-ZcientioRurn, Budap6SHn 1lSis- -de- Rolando-Hotuos-nominatmt SECTIO MATHE~lATICA TOMUS I. G. GRATZER and E. T. SCHMIDT TWO NOTES ON LATTICE-CONGRUENCES

2 TWO NOTES ON LATTICE-CONGRUENCES By G. GRATZER and E. T. SCHMIDT Mathematical Institute, Hungarian Academy of Sciences, Budapest (Received September 13, 1957) In this paper we deal with two questions related to congruence relations of lattices and of abstract algebras. 1. On a problem of G. Birkhoff and O. Frink First of all we define the following notions: A directed set {x a } is a set of elements of a partially ordered set, with the property: to all pairs of elements Xa and xfj of {xa } there exists an xi' in {x a } with X a -<: xl' and xp -<: x y If {x a } is a directed set, VX a exists and VXa = x, then we write X a tx. We call the element X inaccessible from below (t-inaccessible) if X a tx implies X a = x for a suitably chosen X a A lattice L is called weakly-atomic, if to all x < y there exists a pair of elements u, '1", such that x ;'2 u < V ::'S Y and v covers u. Using these notions, we formulate a theorem proved in [2] 1 by G. BIRKHOFF and O. FRINK: The lattice H of all congruence relations on any abstract algebra with finitary operations satisfies the following conditions: (i) H is complete; (ii) X a t X and Yp t Y imply X a n Ypj X n Y (Xa, Yp, X, Y EH) ; (iii) every element is a join of i -inaccessible elements. They ask whether the converse is true or not. We show that the above conditions are also sufficient in those very special cases, when H is a Boolean algebra or a chain. More precisely, we prove: THEOREM 1. If L is a Boolean algebra or a chain, then the following four conditions are equivalent: (1) L is the laffice of all congruence relations of a suitable abstract algebra,. 1 Numbers in brackets refer to the Bibliography given at the end of the paper. 6*

3 84 G. GR.&.TZER AND E. T. SCHMIDT (2) L is the lattice of all congruence relations of a suitable lattice; (3) L is weakly-atomic and complete; (4) the conditions (i), (ii) and (iii) are satisfied in L. REMARK. In case L is a Boolean algebra we may formulate a further condition: (5) L is isomorphic to the atomic complete Boolean algebra of all subsets of a suitable set. PROOF. Case of Boolean algebras. Let B be an atomic complete Boolean algebra. Let us consider the sublattice L of B which consists of the zero of Band all finite joins of the atoms of B, i. e. all the elements a for which [0, a] has a finite length. We state r;;;,ib (the lattice of all congruence relations of L is denoted by e(l». Indeed, L is a relatively complemented, distributive lattice with zero element, and in such a lattice there is a one-one correspondence between ideals and congruence relations. We get such a correspondence, if we let Ie (the kernel of the homomorphism induced bye) correspond to the congruence relation t'1. Now let us consider the correspondence Ie -»V(x; X E Ie). It is routine to check that this is an isomorphism between the lattice f of all ideals of Land B. If we multiply the above correspondences, we get e-»v(x; xele), which is an isomorphism and B. Thus we have proved the implication (5) -» (2). It is well known that (5) is equivalent to (3), see e. g. [1], p By Theorem 3 of [2] the condit: tions (i), (ii) and (iii) imply (3). We know that (1) implies (4), and clearly (1) follows from (2): Thus the proof of this case is completed. Case ofchains. The implication (2)-»(1) is trivial, and (1)-»(4)-»(3) was proved in [2]; thus it is enough to prove that (3) implies (2). Let C be a chain which satisfies (3), and E the subchain of C consisting of all ' t-inaccessible elements of C except of O. Fig. 1 We define the lattice L as follows: the elements of L are the elements x of E, and the elements x' of the dual lattice I, where x' corresponds to x under a fixed dual-isomorphism between E and I; we take an element w, and for all x E E two further elements a." and b,:. Now we define the partial ordering of L: the chains E and E' are partially ordered as before, and for all x, y EE

4 TWO NOTES ON LATTICE-CONGRUENCES 85 we put x' < w < x, ax Ubx x, ax n bx= x' and ax Ua y = xu y,a x n a y = x' ny', lt is easy to check that L is a lattice, see the diagram. We Qd, C. All congruence relations 0 of L are completely determined if we know which elements of E are congruent modulo 0. This is an obvious consequence of the fact that if some of x, x', ax, b x is congruent to some of y, y', ay, by mod 0, then any element in first set is congruent to anyone in the second. further, if x < y, x=y (0) then w= w Uy' = w U(ay nx) w U(a y ny)= =wuay =y(0) is also true, and x y(0) does not imply z=x(@) if x, y < z. Thus there is a one-one correspondence 1] between the congruence relations of L and the ideals of the chain F formed by, E and w (w corresponds to the trivial congruence relation). finally we prove that the lattice of all ideals of the above mentioned chain F is isomorphic to C. 2 Let 1 be an ideal of F. Let rp be the correspondence 1--V(x : x EI), where the complete join is meant in C. The correspondence 1]fjJ is a desired isomorphism between (;<)(L) and C, because each ideal of E is one of the forms 11 {x: x EE, x <: ae E} or 1 2 = {x: x EE, x < a EE}. In the first case a under rp, but in the second case Vx cannot equal to a, for a is an t-inaccessible element. Thus x<a we have proved 0(L) Qd, C, completing the proof of the Theorem Congruence relations of distributive lattices In [3] N. funavama and T. NAKAVAMA have proved that the congruence relations of a lattice form a distributive lattice satisfying the infinite distributive law o n = V (@ n ea). aea ae.4 But they have pointed out to the fact that in e (L) the dual law 0u A 0 a= A (@U(;<)a) aea. aea does not hold in general. The following problem arises: for what lattices L is this dual infinite distributive law of e (L) fulfilled? An other question related to lattice-congruences was proposed by G. BlRKHOFf (see [1], p. 153): PROBLEM 72. find necessary and sufficient conditions on a lattice L, that its congruence relations should form a Boolean algebra. In the paper [4], we have got a sufficient condition for the first question and a necessary and sufficient condition for the second one. from these we 2 This assertion is a special case of KOMATU'S theorem, see [5].

5 86 G. GRATZER AND E. T. SCHMIDT easily obtained the answer of the above mentioned problems in case of distributive lattices. (Problem 72 for distributive lattices was firtly solved by J. HASHIMOTO. For historical notes related to these questions see [4].) Now we work out these problems in a new and common way, which is a more direct and - naturally - a simpler solution than that of [4]. THEOREM 2. The following conditions on the distributive lattice L are equivalent: (a) 0 (L) is a Boolean algebra; (b) the dual infinite distributive law holds in 0 (L); (c) every closed interval of L has a finite length. PROOF. It is well known that (a) implies (b). Indeed, according to a theorem of ]. VON NEUMANN, every complete Boolean algebra satisfies the dual infinite distributive law. Now we verify that (b) implies (c). If in L not every closed interval has a finite length, then there exists a pair of elements a, b (b < a) and a connected chain C between a and b such that C has infinite length. From C one may choose an infinite sequence of intervals [bi' a,] (i= 1,2,...) such that [b,a]=[bo,ao]::j[bllal]::j[b:2,a2]::j... and each [bi, a,] is of infinite length (method of bisection of intervals). We assert that in 0 (C) the dual infinite distributive law does not hold. Let us denote by 0(a, b) the minimal congruence relation in which a _ b. Let 0 ~= V [0 (ao, a,) U0 (bo, bi)]. We show that 0u 1\ 0(ai,b,)=t= 1\ (0u0(aj,b,». i=() It is clear that ao bo(i60 (0 U0(a" bi»); in fact, ao- bo(0u0(ai,b,» for all i, since from definitions ao- ai (0) and bo b, (0), furthermore ai bi (0(ai, b,». So it is enough to prove that ao*' bo(0 u,e 0 (ai, bi»). Now we define a relation on the chain C: Let x y under the relation P if and only if x = y or x =t= y and one of the following conditions is satisfied: 1. bi < x, Y < ai for all i; 2. ai ~ x, y for at least one i; 3. x, y :s b, for at least one i. It is evident that P is a congruence relation. We prove P = (j u 1\ 0 (a" b,).

6 TWO NOTES ON LATTICE-CONGRUllNCllS Obviously, if x (P), then x=y (@ u bi»), therefore P <: IX> U bt). Conversely, x «(:9) is equivalent to x, y E [ai, a] or x, y E [b, be] for some i, i. e. the elements x, y satisfy 2. or 3., that is, (P) and on the other hand, x bi») if and only if x and OJ y satisfy 1. Thus we have (a;, bi) <: P. Summarizing these we get IX> i= 0 IX>,0 (a., bi) ::: P, i. e. u /\ 0 (ai, bi) which was to be proved. At last by the definition of P it is evident that a o b o (P). Applying condition (e) of Theorem 2 of the paper (4), according to which the congruence relations of C may be extended to those on L, so that the.congruence classes in C remain the same, we get that the dual infinite distributive law is not valid (L). Finally we prove that (a) follows from (c). It is obvious that all congruence relations are of the V Bfa, b). Let us consider the <les:o('3).congruence relation $ = V 0 (e, d). " -dj It is trivial that 0 u (/) = 1 and c,f:;d(19j c<d V 0(e,d)= V (0(a,b)n@(e,d» i.e. o:$;d(19) <l"","(19), (l,.. Q a>-o c>-d c $d(19), c>-d 4> is the complement of (9, that (L) is a Boolean algebra. This completes the proof of Theorem 2. Bibliography [I} G. BIRKHOFF, Lattice theory, Am. Math. Soc. ColI. Publ., Vol. XXV, 1948, New York. f2} G. BIRKHOFF and O. FRINK, Representations of lattices by sets, Trans. Am. Math. Soc., 64 (1948), P} N. FUNAYAMA and T. NAKAYAMA, On the distributivity of a lattice of lattice-congruences, Proc. Imp. Acad. Tokyo, 18 (1942), (4] G. GIlATZER and E. T. SCHMIDT, Ideals and congruence relations in lattices, Acta Math. Acad. Sci. Hung., 9 (1958), ] A. KOMATU, On a characterisation of join-homomorphic transformation lattice, Proc. Imp. Acad. Tokyo, 19 (1943),