ISOTONE MAPS AS MAPS OF CONGRUENCES. II. CONCRETE MAPS Ä G. GRATZER, H. LAKSER, AND E.T. SCHMIDT Abstract. Let L be a lattice and let L1, L2 be sublattices of L. Let £ be a congruence relation of L1. We extend £ to L by taking the smallest congruence £ of L containing £. Then we restrict £ to L2, obtaining the congruence £L2 of L2. Thus we have de¯ned a map Con L1 ! Con L2. Obviously, this is an isotone 0-preserving map of the distributive lattice Con L1 into the distributive lattice Con L2. The main result of this paper is the converse in the case that Con L1 and Con L2 are ¯nite. Let L1 and L2 be lattices with ¯nite congruence lattices, and let à : Con L1 ! Con L2 be an isotone map that preserves 0. Then there is a lattice L with L1 and L2 as sublattices such that à is the map Con L1 ! Con L2 obtained by ¯rst extending each congruence relation of L1 to L by minimal extension and then restricting the resulting congruence relation to L2.
1. Introduction Let K and L be lattices and let ' : K ! L be a lattice homomorphism. We then have the associated restriction map rs ' : Con L ! Con K de¯ned by setting x´y for each £ 2 Con L, that is,
((rs ')£)
i®
'x ´ 'y
(£)
rs ' = (' 2 )¡ 1 j Con L ;
where ' 2 : K 2 ! L2 is the map induced by '. Now, rs ' preserves ^ and 1. Observe that rs ' also preserves 0 iff ' is an embedding. We also have a dual concept. We de¯ne the extension of ', xt ' : Con K ! Con L by setting, for each £ 2 Con K, (xt ')£ to be the congruence relation of L generated by the subset ' 2 (£) of L2 : _ (xt ')£ = ( £ L ('x; 'y) j x ´ y (£) ): Date: September 6, 1999. 1991 Mathematics Subject Classi¯cation. Primary 06B10; Secondary 06D05. Key words and phrases. Distributive lattice, congruence, isotone. The research of the ¯rst two authors was supported by the NSERC of Canada. The research of the third author was supported by the Hungarian National Foundation for Scienti¯c Research, under Grant No. T T023186. 1
Ä G. GR ATZE R, H. LAKSER, AND E.T. SCHMIDT
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Now, xt ' preserves _ and 0, and ' is an embedding iff xt ' separates 0, that is, iff (xt ')£ = 0Con L
implies
£ = 0Con K :
In the literature xt is usually denoted Con. We here follow the notation in [1]. We refer to a map that preserves 0 and _ as a f0; _g-homomorphism and to a map that preserves 1 and ^ as a f1; ^g-homomorphism. Note that xt is a covariant functor from the category of lattices with lattice homomorphisms to the category of distributive lattices with f0; _g-homomorphisms as morphisms, and that rs is a contravariant functor from the category of lattices with lattice homomorphisms to the category of distributive lattices with f1; ^g-homomorphisms as morphisms. In Part I of this paper, [1], we proved the following theorem: Theorem 1. Let D1 and D2 be ¯nite distributive lattices, and let à : D1 ! D2
be an isotone map. Then there are ¯nite lattices L1 , L2 , L, a lattice embedding '1 : L1 ! L; a lattice homomorphism '2 : L2 ! L;
and isomorphisms
®1 : D1 ! Con L 1;
®2 : D2 ! Con L2
such that ®2 ± Ã = (rs '2 ) ± (xt '1 ) ± ®1 ; that is, such that the diagram D1 ? ? » = y ®1
Ã
¡¡¡¡! xt '1
rs '2
D2 ? ? » =y ® 2
Con L1 ¡¡¡¡! Con L ¡¡¡¡! Con L2 is commutative. Furthermore, ' 2 is also an embedding iff à preserves 0. In this paper we prove the concrete version of Theorem 1: Theorem 2. Let L1 and L2 be arbitrary lattices with ¯nite congruence lattices Con L1 and Con L2 , respectively, and let à : Con L1 ! Con L2
be an isotone map. Then there is a lattice L with ¯nite congruence lattice, a lattice embedding '1 : L1 ! L; and a homomorphism '2 : L2 ! L such that à = (rs '2 ) ± (xt '1 ): Furthermore, '2 is also an embedding iff à preserves 0. If L1 and L2 are ¯nite, then L can be chosen to be ¯nite and atomistic.
ISO TONE MAPS AS MAPS O F CONGRUENCES. II.
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In Part I, we made extensive use of the technique of \formal inequalities" as formulated in Tischendorf and Tº uma [5] and in Lakser [3]. We could restrict ourselves to considering only \binary" formal inequalities. We announced in Part I that we would consider the concrete version of Theorem 1 in a subsequent paper. What we had in mind was Theorem 2 in the case where L1 and L2 are ¯nite lattices. We envisioned ¯rst applying to the lattices L1 and L2 a result of M. Tischendorf [4] that any ¯nite lattice has a ¯nite atomistic congruence-preserving extension. Any ¯nite atomistic lattice is determined by a set of (not necessarily binary) formal inequalities. We then intended applying the methods of Part I, suitably extended to arbitrary formal inequalities, to derive the above special case of Theorem 2. However, the following theorem in G. GrÄa tzer, H. Lakser, and F. Wehrung [2], with a remarkably simple proof and not relying on the theory of formal inequalities, was discovered after our work on Part I: Theorem 3. Let L0 , L1 , L2 be lattices with ¯nite congruence lattices and let ´1 : L0 ! L1 and ´2 : L0 ! L2 be lattice homomorphisms. Let D be a ¯nite distributive lattice, and, for i 2 f1; 2g, let Ãi : Con Li ! D be f0; _g-homomorphisms such that à 1 ± xt ´1 = à 2 ± xt ´2 :
There is then a lattice L, there are lattice homomorphisms ' i : Li ! L, for i 2 f1; 2g, with ' 1 ± ´1 = ' 2 ± ´ 2 ; and there is an isomorphism ® : Con L ! D such that ® ± xt 'i = Ã i;
for i 2 f1; 2g:
If L1 and L2 are ¯nite, then L can be chosen to be ¯nite and atomistic. This result enables us to prove Theorem 2 rather easily. 2. Results from Part I We recall some results from Part I that we need in order to prove Theorem 2. These results are all very easy, and the reader should be able to verify them without referring to Part I. Let D and D 0 be ¯nite lattices, and let à : D0 ! D be a f0; _g-homomorphism. We de¯ne the M-dual of Ã, ÃM : D ! D0 ;
by setting ÃM x =
_
( y 2 D 0 j à y · x );
for each x 2 D. Dually, if à : D 0 ! D is a f1; ^g-homomorphism, we de¯ne the J-dual of Ã, ÃJ : D ! D 0 ;
by setting ÃJ x =
^ ( y 2 D0 j Ãy ¸ x ):
We then have the following lemmas:
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Ä G. GR ATZE R, H. LAKSER, AND E.T. SCHMIDT
Lemma 1. If à : D0 ! D is a f1; ^g-homomorphism, then à J : D ! D 0 is a f0; _g-homomorphism Lemma 2. If à : D 0 ! D is a f1; ^g-homomorphism, then (à J ) M = Ã.
Lemma 3. If Ã1 : D0 ! D and à 2 : D ! D00 are f0; _g-homomorphisms, then (Ã2 ± Ã1 )M = (à 1 )M ± (à 2 )M : Lemma 4. If ® : D0 ! D is an isomorphism, then ®M = ®¡ 1. Lemma 5. If L1 and L2 are lattices with ¯nite congruence lattices and ' : L1 ! L2 is a homomorphism, then (xt ')M = rs ': 0
Lemma 6. If à : D ! D is a f1; ^g-homomorphism, then à preserves 0 iff à J separates 0. All of the above lemmas have obvious duals|we do not state them since we do not need them in the proof of Theorem 2. The last result we need from Part I is the following: Lemma 7. Let D1 and D2 be ¯nite lattices, and let à : D1 ! D2 be an isotone map. Then there are a ¯nite distributive lattice D, a f0; _g-embedding à 1 : D1 ! D and a f1; ^g-homomorphism
à 20 : D ! D2
with à = Ã20 ± à 1: Outline of proof. We present an outline of the proof because the statement in [1] is slightly weaker; all we stated there about à 1 is that it separates 0. We let D be the set of order-¯lters of D1 , including the empty order-¯lter ?, ordered by the opposite of set containment: F 1 · F 2 iff F 2 µ F1 . We de¯ne à 1 : D1 ! D by setting à 1 x = [x); for each x 2 D1 . Then, clearly, à 1 is an injective map. We de¯ne à 02 : D ! D2 by setting ^ à 20 F = Ã(F ); where Ã(F ) denotes the image of the subset F of D1 under Ã. 3. Proof of Theorem 2 Con L1 and Con L2 are ¯nite and à : Con L1 ! Con L2 is an isotone map. By Lemma 7, there is a ¯nite distributive lattice D, a f0; _g-embedding à 1 : Con L1 ! D, and a f1; ^g-homomorphism à 02 : D ! Con L2 with à = Ã20 ± à 1:
De¯ne à 2 = (à 02 )J : Con L2 ! D:
ISO TONE MAPS AS MAPS O F CONGRUENCES. II.
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Then, by Lemma 2, Ã20 = (à 2 )M : We now apply Theorem 3. We need a third lattice L0 . Set L0 to be the 1-element lattice, and let ´ 1 : L0 ! L1 and ´2 : L0 ! L2 be arbitrary. Since (xt ´i )(Con L0 ) = f0Con Li g for i 2 f1; 2g, we clearly have à 1 ± (xt ´1 ) = à 2 ± (xt ´2 ): Then, by Theorem 3, there are a lattice L, which is ¯nite and atomistic if L1 and L2 are ¯nite, an isomorphism ® : Con L ! D, and lattice homomorphisms 'i : Li ! L, for i 2 f1; 2g, such that ® ± (xt ' i) = à i; for i 2 f1; 2g. Now, compute, using Lemmas 3{5: à = à 02 ± à 1
= (Ã 2 )M ± Ã 1
= (® ± (xt '2 ))M ± ® ± (xt '1 )
= (xt ' 2 )M ± ®M ± ® ± (xt ' 1 ) = (rs '2 ) ± ® ¡1 ± ® ± (xt ' 1 ) = (rs '2 ) ± (xt '1 ):
Observe, ¯nally, that à preserves 0 iff à 02 preserves 0 iff à 2 separates 0 (by Lemma 6) iff xt ' 2 separates 0 iff ' 2 is an embedding, concluding the proof. 4. Concluding remarks In Part I, we also presented two theorems concerning representing a f0; _ghomomorphism and a f1; ^g- homomorphism between ¯nite distributive lattices. The concrete versions are just a special case of Theorem 3. For the record, we state them here. Theorem 4. Let K be a lattice with ¯nite congruence lattice, let D be a ¯nite distributive lattice, and let à : Con K ! D be a f0; _g-homomorphism. Then there is a lattice L, a homomorphism ' : K ! L, and an isomorphism ® : Con L ! D with à = ® ± (xt '): ' is an embedding iff à separates 0. If K is ¯nite, then L can be taken to be ¯nite and atomistic. Proof. In Theorem 3, set L0 = L1 = L2 = K, set à 1 = Ã2 = Ã, and take ´ 1 and ´2 as the identity map.
Ä G. GR ATZE R, H. LAKSER, AND E.T. SCHMIDT
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Theorem 5. Let K be a lattice with ¯nite congruence lattice, let D be a ¯nite distributive lattice, and let à 0 : D ! Con K be a f1; ^g-homomorphism. Then there is a lattice L, a homomorphism ' : K ! L, and an isomorphism ® : D ! Con L with à 0 = (rs ') ± ®:
' is an embedding iff à 0 preserves 0. If K is ¯nite, then L can be taken to be ¯nite and atomistic. Proof. Set à = à 0J , and apply Theorem 4 and Lemmas 1{6 and their duals. References [1] G. GrÄ atzer, H. Lakser, and E. T. Schmidt, Isotone maps as maps of congruences. I. Abstract maps, Acta Math. Acad. Sci. Hungar. 75 (1997), 105{135. [2] G. GrÄ atzer, H. Lakser, and F. Wehrung, Congruence amalgamation of lattices, Acta Sci. Math. (Szeged), to appear. [3] H. Lakser, The Tischendorf-Tº uma characterization of congruence lattices of lattices, manuscript, 1994. [4] M. Tischendorf, The representation problem for algebraic distributive lattices, Fachbereich Mathematik der Technischen Hochschule Darmstadt, Darmstadt, 1992. [5] M. Tischendorf and J. Tº u ma, The characterization of congruence lattices of lattices, manuscript, 1993. Department of Mathematics, University of Manitoba, Winnipeg MB R3T 2N2, Canada E-mail address, G. GrÄ atzer:
[email protected] URL, G. GrÄ atzer: http://www.maths.umanitoba.ca/homepages/gratzer/ E-mail address, H. Lakser:
[email protected] Mathematical Institute of the Technical University of Budapest, M} uegyetem rkp. 3, H-1521 Budapest, Hungary E-mail address, E.T. Schmidt:
[email protected] URL, E.T. Schmidt: http://www.bme.math/~schmidt/
