Nieminen, Juhani

English

Nieminen Juhani

University of Szeged

Acta Sei. Math., 38 (1976), 359—341' Derivations and translations on lattices JUHANI NIEMINEN 1. Introduction. Let S be a meet-semilattice and (p a single-valued mapping of S into itself, (p is called a meet-translation on S [3], if <p(xAy)=<p(x)Ay for each pair of elements x, If S—L is a. lattice and (p a single-valued mapping of L into L such that \ i <p(xVy) = <p(x)V<p(y) . a n d (p(xAy) = (<p(x)Ay)\/(q>(y)Ax) for each pair x, y£L, then (p is called a derivation on L [5]. As shown by SZASZ in [5], a single-valued mapping on a lattice L is a derivation on L if and only if it is a meet-translation as well as an endomorphism on L. Each meet-translation cp on S has the following properties [3]: q>(:C)SJC, <p(<p(x))=(p(x), and x^y=xp(x)^(p(y). Moreover, in a lattice L the fixedelements of (p, i.e. the elements t=(p(t), constitute an ideal of L [4]. As shown in [4], K^ determines (p uniquely. ; In this note we shall illuminate the dependence of q> from the properties of the ideal Kq. A single-valued mapping (p of a join-semilattice V into itself is called a. join-translation on V, if <p(xVy)=(p(x)Vy for each pair x, y€ V. The results on translations in the papers [1]—[4] are given in terms of join-translations. As we shall consider here meet-translations, we always use the dual of the corresponding result obtained in the papers [1]—[4]. 2. Derivations on lattices. We denote by; . / ( £ ) the lattice of all ideals of a lattice Z,; (z] = Sz , x, z£L}. T h e o r e m 1. An ideal I of a lattice L generates a meet-translation (p on L such that I=K']=(<?>(>')] for each y^L. Received September 25,1975. 360 J. Nieminen Conversely, let /A(.y]=(&] for each y£L. We put q>(y)=k and show that q> is a meet-translation on L. Obviously q> is single-valued and K^—l. IA(xAy\ = =(/A(JC])A(J>]; thus <p(xAy) = tp(x)Ay and the theorem fol lows. T h e o r e m 2. Let D be an ideal of a lattice L generating a meet-translation q> on L. Then (p is a derivation on L if and only if DA((y]V(x])=(DA(y])V(DA(x])for each pair of elements x, y£L. Proof . As D generates a meet-translation (p on L, DA(y]=(k] for each y£L. Let the condition of the theorem be valid for the elements x,y£L. Then D A ( J C V > ' ] = (I>A(JC])V(Z>AO']), whence <p(x\/y) = <p(x)\l<p(y). Furthermore, DA(xAy]=(DA(x])A(J]=(DA(y])A(*]={(/)A(X])AFJ]}V{(DA(J])A(*]} which im-plies that (p(xAy)=(cp(x)Ay)W(<p(j)Ax). Conversely, let (x\Jy)=q>(x)V(p(j) implies that KVA (xVy]=(<p (XVJ) ]= (<p (x)]V(cp ( J ) ] = ( * , A ( X ] ) V (Kv (y]). This completes the proof. A n element x o f a lattice L is called distributive, if xi\(y\J z)=(xAy)\/(xAz) for each pair y, z£L. The following lemma shows that the condition of Theorem 2 reduces to the distributivity of D in the lattice J(L). , L e m m a 1. Let T be an ideal of a lattice L such that 77\((;c]VO]) = =(TA(x])\J(T/\(y]) for each two elements x,y£L. Then TA(7VK)=(TM)\/(Tf\K) for each two elements J, K^J(L). Proof . As is well known, it is sufficient to show that TA(I\l K)Q(TAI)\I (TAK) Let xeTA(I\/K), i.e. x£T and x^iVk for some / € / and k^K. Then (x]g(/]V(/c] and *€(>] = TA(JC] Q(TA(/]) V (TA (k])I (TA/)V ( T A K ) , and the lemma follows. The lattice J(L) of a modular lattice L is modular. Already the relation TA(/VK)=(77\/)V(TAK) implies the neutrality of T i n a modular lattice [6, Thm. 103 and its corollary]. So we can write C o r o l l a r y 1. A meet-translation <p on a modular lattice L is a derivation on L if and only if Ky is a neutral element of the lattice ¿(L). By the join of two derivations (p and A on a lattice L we mean the mapping <p(x)\l k(x) o n L and by the meet the mapping cp(x)Ak(x). In the fo l lowing we con-sider some conditions under which the join and meet defined above are also de-rivations on L. T h e o r e m 3. The meet of two derivations (p and k on a lattice L is always a derivation on L. Moreover, the join of <p and k is a derivation on L if Kv and Kx are neutral ideals of L. Proof . ( ^ A ^ ) A W = ( ^ A ( x ] ) A ( ^ A ( x ] ) = ( ^ ( x ) A A ( x ) ] and so Kkl\K, gen-erates a meet-translation which is <p(x)Ak(x). Further, (K<pAKx)A(x\/ y] = Derivations and translations on lattices 361 =-K9A{A";lA(xV>']}, and by applying now Kx and sequently, (K<pAKx)A(x\/y] = = w h e n c e (p(x\Jy)AA(x\J y)=((p (x )AA (x ) ) A V((p(y)AA(y)). This means that the meet of A and <p is a join-endomorphism, too, and the first assertion follows. Let the ideals Kv and Kx be neutral and let us consider the ideal K^M Kx. V IS:^) A ( * ] A C * ] ) V ( J C 4 A ( * ] ) = ( 9 » ( * ) ] V ( A ( J C ) ] = ( 9 » (x)V A (*)]. Thus the ideal KvVKx generates a meet-translation /?(x)=A(x)V<p(x) on L. The join of two neutral ideals is also a neutral ideal, and so (K^V Kx) A (xV j ] = {(*:„ V Kx) A (x]} V {(A; V Kx) A A (.)>]}• Hence fS(x) is a join-endomorphism on L and also a derivation on L. In [5, Thm. 3] SZASZ has shown that the product <pA of two derivations on a lattice L is always a derivation, and moreover, (pX (x)=q> (A (x))=cp(x)A A (x). As shown by SZASZ [ 5 , Thm. 2 ] , the derivations of a lattice L are exactly those meet-translations of L that are also endomorphisms on L. As immediate corollary of the construction of KOUBIAR in [ 1 , Thm. 1 ] , we can write T h e o r e m 4. On a modular lattice L there is a one-to-one correspondence between meet-translations <p and congruence relations d9 having the property (i) There is in L a neutral ideal T such that every rest class modulo 9^ contains exactly one element o f T . The congruence relation 9P relating to the meet-translation (p and the meet-translation (pe relating to the congruence relation 9^ are characterized by (ii) and (in), respectively: (ii) x9q,y<=xp(x)=(p(y), x,y£L; (iii) (pe ( x ) = x " £ T for which x9B x". Now we can prove an extension of [2, Thm. 1] T h e o r e m 5. Let L be a modular lattice. The set of all congruence relations 9^ relating to the derivations (p on L constitutes a sublattice of the lattice 9(L) of all congruence relations on L. Proof . According to Theorem 4, x9vyo(x]AKV =(j>]AK^ for each derivation (p on L. As L is modular, for each derivation (ponL the ideal is a neutral element of J(L) (Corollary 1). Hence, for any two derivations ip and A on L the mappings <p(x)VA(x) and <p(x)AA(x) are derivations on L, too (Theorem 3). Let P(x) = =<p(x)AA(x). We prove 9^=9^ 9X by showing that 1) e v M B ^ B f , and 2) 9^9^ 1) x d v y ^ A K ^ W A K ^ i x M K v A K J ^ W A i K v A K J ^ x d e y , and so 0 9 = 0 f . Similarly we see that 9 X ^9 P , whence 9 9 \ / 9 x ^9 f i . 2) Le t x9i>yo(x]AKq>AKx=(y]AK<pAKxoxA(p(x)AX(x)=yA<p(y)AA(y). O n the other hand, xA<p(x)9xxA<p(x)AX(x), and moreover, x9vxA<p(x). Hence, 362 J.'Niemineri x(0„V0x)xA<p(x)AA(x). Similarly we see that y(09V0JyA<p(y)AX(y), and-by combining these results we obtain x(d<fl\ldx)y. Thus 69\/dx^dfi. Let a(x)=(p(x)Vl(x); we prove that 0a=0„A0X by showing that 3) 0„S0„Al9i and 4) 0 a s0^A0 A . 3) Let x(OvAQdy*>xQ9y&n&xeiy^(x)AKv=(y]AK9s.n<l(x]AK l i=(y]AKk=> ( x ] A ( K 9 V K X ) = ( y ] A ( K p V K ) ) o x Q a y . Thus 0 a ^ 0 9 A 0 , . 4) Let x0 a > > ~ ( x ] A ( V K x ) = ( y ] A ( K (x]A(K<pVKx)AK<l,=(x]AK, =' =(j>] A (K^M K^A K9=(y]AK9, and so x09y.. Similarly we set that x6xy, too. Con-! seqiiently, xid^Ad^y, which implies the desired result. i . , A meet-translation ((y) for each two elements x , y £ L . T h e o r e m 6 . Let M be an ideal of a lattice L generating a meet-translation <p on L. Then <p is a weak derivation on L if and only if MA((x]\/(y])=(MA(x]) V V(MA(y])for each two elements x,y£L and x £ M . The proof follows the lines of that of Theorem 2, and hence we omit it. Further, the proof of the following lemma is analogous to that of Lemma 1, and hence it is omitted. . \ L e m m a 2. Let T be an ideal of a lattice L such that TA((x] V ( j ] ) = ( 7 7 \ ( x ] ) V M(TA(y\) for each two elements x,y£L, x£T. Then TA(I\/K) = (TAI)\J(TAK) for each two elements I, K£J(L), IQ T. • • As shown by SzAsz [4, Thms. 4 and 5], the distributivity and modularity of a lattice L can be characterized by derivations and weak derivations of L, respectively. It is interesting to see that these characterizations reduce the distributivity;(the modularity) of L to the distributivity (the modularity) of «/(£), as one can deduce from Theorem 2 and Lemma 1, and from Theorem 6 and Lemma 2, respectively. 3. Meet-translations on meet-semilattices. In this section we shall show a connec-tion between meet-translations on meet-semilattices and lattices. We shall consider' meet-semilattices only, and hence we shall use the brief expression semilattice in-stead of meet-semilattice. Note that in 5 a nonvoid set / is an ideal if (i) x £ / and r f e x imply and (ii) x,y£l imply xAy£I. S is up-directed if for each pair x, y£ S there is an element k£S such that k^x, y. In particular, if S is up-directed, then IA J is an ideal of S for each two ideals I and J of S. .¡/.'•v. T h e o r e m 7. Let S be an up-directed semilattice and cp a meet-translation on S. Then (p generates a meet-translation <pe on the lattice ./(£) of all ideals of S defined as follows: <p9(I)={x\x^<p(y);yO€S(S)}. Proof . At first we show that (pB(I) is an ideal of S. Let x£<p9(7) a n d / S x . ' Then there exists an y£I such that raxs<pO>), and so r£<pg(I). Let a,b£cpg(I). Thus aAb^(p(ya)A(p(yb)=(p(yaAyb), where yaAyb£l\ therefore aAbe<p9(I): ••<;> J. Nieminen: Derivations and translations on lattices 363-Clearly q>g is a single-valued mapping on thus it remains to show that q>'(IAJ)=(pe(I)AJ. Let x£(pg(IAJ). Then there is an element y^IAJ such that On the other hand, y^iAj with some / € / and j£J, and (p(y)^cp(iAj)= =<f>(i)Aj. Thus x^q>(i)Aj with (p(i)f_(pg(I) and j£J, whence x£<p9(I)AJ. This shows that (pB(IAJ)Qcpa(I)AJ. Let now x€9®(/)AJ. Then x S r A j for some r£<p9(I) and j(LJ. Furthermore, there exists an such that r^<p(i), and so x^(p(i)A j=tp(iA j), where iAj£IAJ. Therefore, x£<pg(IAJ), and the relation (pg(r)AJ^(p^(IAJ) holds. Consequently, <pg(IAJ)—(pB(I)AJ, and the theorem follows. Let [z)= x, z£S} . The validity of the following assertion is obvious. T h e o r e m 8. A meet-translation (p on S(S) is generated by a meet-translation X on S, i.e. (p=X9, if and only if for each x£S there is an element k£S such that <p([x))=[k). Acknowledgement . The author wishes to thank the referee for his valuable comments and suggestions. I i References [11 M. KOLIBIAR, Bemerkungen über Translationen der Verbände, Acta Fac. R. N. Univ. Comenianane Math., 5 (1961), 455—458. [2] J. NIEMINEN, A note on relations of distributive lattices, Acta Fac. R. N. Univ. Comenianae Math., to appear. , [3] G. SZÁSZ, Die Translationen der Halbverbände, Acta Sei. Math., 17 (1956), 165—169. [4] G. SZÁSZ, Translationen der Verbände, Acta Fac: R. N. Univ. Comenianae Math., 5 (1961),. 449—453. I [5] G. SZÁSZ, Derivations of lattices, Acta Sei. Math., 36 (1975), 149—154. [6] G. SZÁSZ, Introduction to lattice theory. Academic. Press and Akadémiai Kiadó (New York— London and Budapest, 1963). DEPARTMENT OF TECHNICAL SCIENCES, FINNISH ACADEMY LAUTTASAARENTIE 1 00200 HELSINKI 20, FINLAND

Derivations and translations on lattices

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Derivations and translations on lattices

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