Cornish, William H.

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Mathematica SlovacaWilliam H. CornishOn the Chinese remainder theorem of H. DraškovičováMathematica Slovaca, Vol. 27 (1977), No. 3, 213--220Persistent URL: http://dml.cz/dmlcz/131801Terms of use:© Mathematical Institute of the Slovak Academy of Sciences, 1977Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document must containthese Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.czMoth. Slovaca 27,1977, No. 3, 213—220 ON THE CHINESE REMAINDER THEOREM OF H. DRASKOVICOVA WILLIAM H. CORNISH Introduction. As a by-product of her investigation into strongly permutable sets of congruences, H. Draskovicova [4; Corollary 3.4] proved that an upper subsemilattice of the lattice of congruences of a universal algebra satisfied the Chinese remainder theorem if and only if the sublattice generated by the subsemilattice is not only distributive but also consists of permuting congruences. The purpose of this note is twofold. Firstly, we give a proof of Draskovicova's theorem that is based on B. Jonsson's theorem which gives an elegant necessary and sufficient condition for the sublattice generated by a subset of a modular lattice to be distributive. Since R. Balbes [1] has given an elementary proof of Jonsson's theorem, this means that Draskovicova's theorem is independent of the Axiom of Choice. Secondly, we consider some naturally arising instances of this Chinese remainder theorem. Our main examples involve congruence-kernels in a distribu-tive pseudocomplemented lattice and the so-called distributive ideals of a general lattice. We precede these examples with a brief discussion of ideals (con-gruence-kernels) in a universal algebra. 1. The Chinese Remainder Theorem. The notation and terminology of this article is based on that of [5]. A set ^ of congruences on a universal algebra 21 is said to satisfy the Chinese remainder theorem (is finitely strongly permutable, in the terminology of [4]) if, for any 0U 02, ...,dn 6 0* and any au a2, ...,aneA, 4-S4(0,v0/)forany/, /=-l ,2, ..., n implies that there exists aeA, such that a = 0/(0/) for each j=l,2,...,n. In [5; Chapter 5* Exer. 68] G. Gratzer generalized the Chinese remainder theorem of Zariski and Samuel [12; Theorem 18, p. 280] to universal algebras. The proof is via finite induction and can be extended slightly to establish the following lemma which has already been observed by DraSkovidova [4; Corollary 3.3]. 213 Lemma 1.1. Let ££ be a sublattice of the lattice of congruences of a universal algebra ?l. Then, J£ satisfies the Chinese remainder theorem if and only if if is distributive and consists of permuting congruences. The next lemma is due to B. Jons son [9]; a proof is also presented in the book [11; Theorem 34, p. 93]. Recently an elementary proof was given by R. 3albes Lemma 1.2. Let H be a non-empty subset of a modular lattice L. The sublattice ofL, which is generated by H, is distributive if and only if, for any xx, x2,..., xn and yi, y2, . . , ym in H n tn tn / n A*, A Vy, = V I A*. * M i=\ j=\ j=\ X i = l We now come to Draskovicova's theorem which followed as a consequence of her detailed examination of permutability notions. Moreover, Examples 3.1 and 4.1 of [4] show that in a certain sense the result is the best that can be hoped for. Theorem 1.3. Let 9 be an upper subsemilattice of the lattice ^(Sl) of a universal algebra SI. Then, SP satisfies the Chinese remainder theorem if and only if the sublattice L(<3>) of <€(%), generated by 0>, is distributive and consists of permuting congruences. Proof. Because of 1.1 it is sufficient to verify the necessity of the conditions on L($P). To do this we first prove the following: (1.1) UWx,W2,...,Wn\ &x,<P2,...,<Pme2Pthen n m n m nn(^v<p /)=n^on<p,. i=\ y = l i = l j=\ Suppose that x and y are elements of A which are congruent with respect to the left-hand side of (1.1). Let 6l=Vx, ..., 6n = Wn, 0n+l = 4>l9 ..., dn+m = 0m and ax = ... = an =x, while an+x = ... = an+m =y. Then, a, = ay(0, v0y) for any /, / = 1, 2, ..., n + m. As 2P satisfies the Chinese remainder theorem, there exists a eA such that a = aj(0j) for all / = 1, 2, ..., n + m. It follows that x is congruent to y with respect to the product on the right-hand side of (1.1). It is now clear that 1.1 holds. Since 3P is an upper subsemilattice of ^(Sl), (1.1) immediately implies (1.2) L(&) = \C]di:du62,...,dne0>-, AZ^I} and L(@>) consists of permuting congruences. Hence, (1.3) L(P) is a modular lattice. We now establish 214 (1.4) For any Wx, W2,..., Vne&and W, 0e&, (VxnV2n...n¥n)n(Wv0) = (Vxn...nVnnV)o(Vxn...nVnn0). Suppose that x and y are elements of A which are congruent with respect to the left-hand side of (1.4). Let 0X = WX, ..., dn = Wn, 0n+x = <P, 0n+2=0 and ax = ... = an=an+x=x while a„+2 = y. Then, a, = ay(0, v07) for each i,j=\, ..., AZ + 1, an+x = an+2 (0n+xv0n+2), and ai = an+2(0iV0n+2) for all i = l, ..., AZ, since * — y(#) f° r e a c h /= 1, 2, ..., AZ. That is, a, = a;(ft v07) for all / , / = 1, ..., n + 2. Hence, there exists aeA such that a = a7(0y) for /= 1, ..., n + 2. Then, a = x(W{) for # = 1, ..., /z, a = x(*P) and a = y(0). But ^ = y(V>;) for / = 1 , ..., AZ and so a==y(Vl) for /= 1, ..., /z. Hence, x is congruent to y under the product on the right-hand side of (1.4), and it follows that (1.4) holds. Since & is an upper semilattice, induction on (1.4) yields (1.5) For any Wx,..., Vn,0x,..., <Pme<? / = i j=\ j=\ > i = \ J The theorem now follows from (1.1), (1.2), (1.3), (1.5) and Lemma 1.2. 2. Ideals and Examples. Let z be a fixed element of a universal algebra. For 0 in ^(21), define z-ker 6 = {aeA: a = z(0)} and call it the z-kernel of 0. A z-ideal or con-gruence ~z-kernel is a (non-empty) subset J of A such that /=z-ker 0 for some congruence 0 in ^(21). In practice the element z is usually identified with a nullary operation in 21. Either for the sake of brevity or because there is no ambiguity the prefix "z-" is generally omitted from the above. Thus, in what follows we will speak of "ideals" and denote the collection of all ideals by /(2l). Ideals in this general context are discussed in [7; 0.2.39—0.2.52, p. 78—82] and a fundamental result is [7; Theorem 0.2.41, p. 78] which we single out in the next lemma. Lemma 2.1. Ordered by set-inclusion, the collection /(2l) of ideals is an algebraic closure system on A and hence a complete compactly generated lattice. For an ideal / , let 0(J) be the smallest congruence on 21 which has / as its kernel. Of course, 0(J) = n { 0 e ^(21): ker 0 = J) and the map /.-» 0(J) is an order-emb-edding of /(2l) into ^(21). In general, only the following property of this map would seem to hold. Proposition 2.2. For any collection {/A} of ideals, 0(v/A) = v0(/A). In particul-ar, if {/A} is directed then 0( v/A) = 0(u/A) = u0(/A). 215 ' Proof. For / G / ( 9 1 ) , let d[J]=v{d(a, b): a, be J}. Thus, 0[/] is the smallest congruence on 91 identifying all the elements of / . Hence, 0[/]cz0(/). But this inclusion implies ker (0[ / ] )c / and so ker(0[/]) = / . Hence, 0[/] = 0(/). For / , K6/(91), /czK certainly implies 0[/]cz0[K]. Hence, /czK if only if 6(J) cz d(K). We are now able to prove the assertion. For if Jft e {/A} then /„ cz v/A and so V0(/A)cz0(v/A), while/„ czker(v 0(/A)) so that v/A czker(v0(/A)). Hence, 0( v/A) c 0 (ker( v 0(/A))cz v 0(/A). Hence, 0( v/A) = v 0(/A). The rest follows from the algebraic nature of the closure systems involved. Of course, we have no right to expect that 6(JnK) = 0(J)n0(K) for any two ideals / and K. Thus, in general, {0(/): J e /(9l)} is only an upper subsemilattice of #(91). Of course, it does happen when 91 is (z-) weakly regular, i.e. when the map Jt->d(J) is a bijection and so induces a lattice isomorphism of /(9l) onto #(91), whose inverse is the map 0>-»ker0 of #(91) onto /(9l). This is the case for groups, rings, Boolean algebras etc. However, there is another case which is not without interest. Theorem 2.3. Let ^ be a distributive sublattice ofJ(%). Assume also that SP has the same smallest element and largest element as /(9l) and 5P is closed under the formation of arbitrary suprema. Let P(3P) be the set of meet-irreducibles of 2P. Then the following two conditions are equivalent. (i) For any / , Ke&, 6(JnK) = d(J)nd(K). (ii) For any J e®, 0(/) = n{0(P): /czP, PeP(^)}. Proof. (i)=>(ii). Clearly, 0(/)<=n{0(P): /czP, PeP(SP)} for any Je&. We prove the opposite inequality by a contrapositive argument. That is, suppose that a, beA and a^b(d(J)). Consider the set 3f{={Ke&: /czK, a^b(d(K))}, partially ordered by set-inclusion. Let # be a chain in JC. By the hypotheses on £r\ u { C e ^ : Ce%}e&. From Proposition 2.2, this ideal is actually in X. Thus, Zorn's lemma implies that JC has maximal elements. Let Q be such an element. It remains to prove that Q is meet-irreducible in &. Firstly, notice that since a^b(d(Q)), Q is not A, the largest element of ^ . Let A, B e SP and assume that A£Q and B£Q. Then, QvA and QvB both properly contain Q. Hence, a = b(0(QvA)nd(QvB)). By (i) and the distributivity of ^ , it follows that a = b(6(Qv(AnB))). Hence, A nB<£ Q. The distributivity of ^ ensures that Q is meet-irreducible. (ii)4>(i) is easy due to the distributivity of ^ . Indeed, 0(/nAT)cz0(/)n0(K) holds for any / , Ke/(9l). Now, let Pe£P be meet-irreducible and suppose JnK^P for given J,Ke@. Then, /czP or KczP. That is, 0(/)cz0(P) or 0CK)cz0(P). In either case 0(/)n0(K)cz0(P). Thus, (ii) implies 0(/)n0(lOcz0(/ni<:). Taking ^ = /(9l), z = 0, and 91 as a distributive lattice with 0, considered as an algebra (A ; v, A, 0) of type (2,2,0) , Theorem 2.3 can be applied to obtain a new 216 proof of [3; Lemma 1.2 (ii)]. It can even be applied to the lattice & of standard ideals of a general lattice with 0. This is because of [6; Theorem 3, p. 33]; for the sake of brevity, we will omit the details and implications of this application. It can also be used to give new interpretations of the pathological behaviour of semirings as noted in [2] and an explanation of some of the conditions, in particular (x), of [3; Theorem 3.2], wherein Stone lattices are characterized by the behaviour of their congruence kernels in the variety of distributive pseudocomplemented algebras. Let us look a little more closely at these two instances, particularly in relation to Theorem 1.3. By a semiring, we mean an algebra 21 = (A ; + , •, 0) of type (2, 2, 0) such that (A ; +, 0) is a commutative semigroup with 0 as its zero, (A, •) is a semigroup and the multiplication " •" distributes over addition from either side and a • 0 = 0. Thus, we consider semirings as generalizations of both rings and distributive lattices. Also, we take z = 0. It is easily checked that a non-empty subset / of A is an ideal if and only if (2.1) a, beJ imply a + beJ, (2.2) aeJ, be A imply ab, baeJ, and (2.3) a + beJ and aeJ imply beJ hold simultaneously. Indeed, to verify this we also need to verify that 8(J) is given by % (2.4) x = y(6(J))(x, ye A) if and only if x + a = y + b for some suitable a, be J. In general we have no right to assume the permutability of the 6(J) nor to assume the intersection formula: 0(JnK) = 0(J)nd(K). In [2], a 5-element semiring in which both operations + and • are commutative and idempotent is given. The intersection formula failed but all the congruences on 21 are in the sublattice generated by the subsemilattice {6(J): Je /(2l)} and what is more ^(21) is distributive and consists of permuting congruences. Thus, we get a particular case of Theorem 1.3, where the fact that we are dealing only with an upper semilattice of ^(21) was of prime importance to [2]. Now consider a distributive pseudocomplemented lattice considered as an algebra 21 = (A ; v, A, *, 0,1) of type (2 ,2 ,1 ,0 ,0) . Take z = 0. Then, the ideals of 21 are examined in detail in [3]. We will use the notation and terminology of [3] freely, except that we will speak of ideals in place of congruence-kernels of •-congruences. There the intersection formulae of Theorem 2.3 failed badly, but after all that was the interest in the paper, c.f. [3; Theorem 2.5 and Theorem 3.2, (x), (xi) and (xii)]. However, for /e / (2t ) and O^n^co, each of the congruences 2rt(/) (see [3; Theorems 1.5, 1.6, 1.7 and 2.5]) is determined by the minimal congruence identifying some filter on A. And the minimal congruences W(F) associated with filters F of a distributive lattice form a distributive sublattice of #(21) which consists of permuting congruences c.f. the "dual" of Theorem 2.3 and 217 [3; Lemma 1.2, Lemma 1.3] (N.B. The statement of [3; Lemma 1.3] should be ended by the clause "if they are congruence relations"). Thus, we can summarize and apply Theorem 1.3 to obtain Theorem 2.4. Let 21 = (A ; v, A , *, 0, I) be a distributive pseudocomplemented lattice considered as an algebra of type (2, 2, 1, 0, 0). With the notation of [3], O^n^co and 0>n=(In(J): /e /(2l)) , SPn is an upper subsemilattice of <ti($l) satisfying the Chinese remainder theorem and consequently generates a distribu-tive sublattice of permuting congruences in ^(21). When tz = 0, SPn is actually a sublattice of ^(21), while for l^rz^cO, 2Pn is a sublattice if and only if 21 is a Stone lattice. This is an easy consequence of our previous remarks and [3; Theorems 2.10 and 3.2 (xii)]. Finally, let us turn to the so-called distributive ideals which can be defined as those ideals / of a lattice which satisfy either of the equivalent conditions of the following lemma. The details are omitted; for a comparison with standard ideals, see [6]. Lemma 2.5. For a (lattice)-ideal J of a lattice 21 = (A ; v, A), the following conditions are equivalent. (i) / v (HnK) = (JvH)n(JvK) for any (lattice)-ideals H and K (ii) The relation 0(J) defined by x = y(d(J))(x, ye A) if and only if xvj = yvj for some I eJ is a congruence relation on 21. (Note: 6(J) can be defined equivalently by x = y(0(J))(x, ye A) if and only if Cr]v/=(y]v/ ) . Of course, here we have not insisted on the existence of OeA (i.e. a disting-uished element z) and 6(J) of Lemma 2.5 is the smallest congruence on 21 having the distributive ideal / as a class. The collection />(2I) of distributive ideals of 21 is a complete upper subsemilattice of the lattice of ideals of lattice 21 and ^(Z?(2l)) = {#(/): JeD(%)} is a complete upper subsemilattice of <€(&) (the proof is similar to that of Proposition 2.2). Theorem 2.6. Let 21 = (A ; v, A ) be a lattice. Then the upper subsemilattice SP(D(VL)) of ^(21) and consisting of the minimal congruences associated with the set D(W) of distributive ideals satisfies the Chinese remainder theorem. Proof. Let Ju /2, ..., /„eD(2l) and xu x2, ..., xneA be such that Xi=Xj(d(Ji)vd(Jj) = d(Jtv Jj)) for each /, / = 1, 2, ..., n. It follows that there exist kteJi and £, eJ; such that x{v ktv k},=xivk{vk, for each /, I = l, ..., n. Hence, we certainly have (xivki]vJi = (xivki]vJi for each /, I = l, ..., n. Let x = (x1vk1)A...A(xnvkn). Then, (x]vJi = ((x1vk1]n...n(xnvkn])vJi = ((x1vk1]vJi)n...n((xivki]vJi)n...n((xnvkn]vJi) 218 = ((x,vki]vJj)n...n((Xj]vJj)n...n((xjVkn]vJI) = (x /]v/ /, using the distributivity of /,. That is, X = XJ(6(JJ)) for / = 1, ..., n. In general, the intersection JnK of two distributive ideals / and K may even turn out to be distributive and yet 0(JnK) £ 6(J)nd(K). This is illustrated by the five element non-modular lattice {0, a, b,c,l: 0 < a < l ; 0 < c < b < l } , wherein (a]n(b] = (Q], (a], (b] and (0] are all distributive ideals, yet b±c and b = c(6((a])n0((b])). On the other hand Theorem 2.6 and Theorem 1.3 yield positive results. Corollary 2.7. Let / , , . . . , / „ , K.,..., Km be n + m distributive ideals in a lattice 21. Then, the congruences f]6(Jr) and f\d(Js) are permutable. r = l s=l Corollary 2.8. Let 21, 21,, 2l2, ..., 2l„ be lattices with 0. 777en, 21 is isomorphic to the direct product of 21,, 2I2,..., 2l„ if and only if there exist distributive ideals /,, /2, ..., /„ in 21 5i/cA tAat (i) the Jt are pairwise comaximal i.e. r±s implies JrvJs = A, (ii) f)d(Ji) = o), and (iii) for each i=l, ..., n, % ^Wd^). Proof. Corollary 2.8 follows from Corollary 2.7 using a familiar argument, see, for example [10; Corollary 1.6, p. 67]. We note that Corollary 2.7 simplifies for standard ideals ([6; Theorem 3, p. 33]) and takes into account many known results, see [8; Theorem 2 and its corollaries]. The ideals / of Corollary 2.8 are easily seen to be standard when 21 = 21, x ... x 2I„ U, = {fe A: /(/) = 0} and 0(Jt) is given by f=g(f,geA) if and only if /(/) = g(i) or equivalently either fvk = gvk or fvg = (f/\g)vk for some keJt) and hence, an alternative form of Corollary 2.8 can be stated, wherein the ideals Jt are n assumed to be standard and (ii) is replaced by (ii)' p | / , = (0]. REFERENCES [1] BALBES, R.: A note on distributive sublattices of a modulaг lattice. Fund. Math. 65, 1969, 219—222. [2] CORNISH, W. H.: Subdiгectly irгeducible semirings and semigroups without nonzeгo nilpotents. Canad. Math. Bull. 16, 1973, 45—47. [3] CORNISH, W. H.: Congruences on distributive pseudocomplemented lattices. Bull. Austral. Math. Soc. 8, 1973, 161—179. [4] DRAŠKOVIČOVÁ, H.: Permutability, distributivity of equivalence relations and direct products. Mat. Čas. 23. 1973. 69—87. 219 [5] GRÄTZER, G.: Universal algebгa. Princeton 1968. [6] GRÄTZER, G., SCHMIDT, E. T.: Standard ideals in lattices. Acta Math. Acad. Sci. Hungaг. 12, 1961, 17—86. [7] HENKIN, L., MONK, J. D., TARSКI, A.: Cylindric algebras. Paгt I. Amsterdarn 1971. [8] IQBALUNISSA: Permutable congruences in a lattice. Illinois J. Math. 10, 1966, 235—239. [9] JÓNSSON, B.: Distгibutive sublattices of a modular lattice. Proc. Amer. Math. Soc. 6, 1955, 682—688. [10] PIERCE, R. S.: Introduction to the theory òf abstгact algebras. New Yoгk 1968. [11] SZÁSZ, G.: Introduction to lattice theoгy. New York 1963. [12] ZARISKI, O., SAMUEL, P.: Commutative algebra. Vol. 1. Princeton 1958. Received January 13, 1975 Department of Mathematics Dalhousie University Halifax, N.S., Canada Permanent address: School of Mathematical Science The Flinders University of South Australia Bedford Park, 5042, S.A. Australia 220 

On the Chinese remainder theorem of H. Draškovičová

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On the Chinese remainder theorem of H. Draškovičová

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Institute of Mathematics AS CR, v. v. i.