General Lattice Theory: 1979 Problem Update

Transcription

1 Algebra Universalis, 11 (1980) Birkhauser Verlag, Basel General Lattice Theory: 1979 Problem Update G. GRATZER Listed below are all the solutions or partial solutions to problems in the book General Lattice Theory known to the author by the end of Since the manuscript of the book has been in rather wide circulation for about six years and the number of papers that have a bearing on the problems is rather large, it was felt that such a list would be a useful aid for further research. Chapter I. 7. A. P. Huhn pointed out to the author that for any nontrivial equational class K of lattices. the variety generated by the sublattices of members of K is the variety L of all lattices. Indeed. for a set X. form the lattice A of all finite subsets of X. Then Sub (A) contains the lattice of all finite partitions of X, hence references to Corollary IVA.3 and Corollary IVA.5 complete the proof. Thus there are no proper equational classes of lattices closed under the formation of sublattice lattices. 13. D. Kelly [1] proved that any natural number is the dimension of a suitable finite planar poset. 22. Some of the papers yielding a characterization of transferability have now appeared or have been submitted for publication. See G. Gratzer and C. R. Platt [1], G. Gratzer. C. R. Platt. and B. Sands [1]. C. R Platt [1]. 23. From the characterization of finite transferable lattices. it is obvious that the answer is in the affirmative: any sublattice of a transferable lattice is transferable again and the dual of a transferable lattice is also transferable. 24. Sharply transferable lattices are characterized in G. Gratzer and C. R. Platt [1]. solving this problem. 25. The problem asks for a direct proof that a sublattice and the dual of a sharply transferable lattice is sharply transferable again. The proof in A. Day [2] is simple Presented hy W. Taylor. Received Decemher O XO. I<)79. Accepted for puhlication in final form May 396

2 Vol 11, 1980 General Lattice Theory: 1979 Problem Update 397 but it does not qualify as a direct proof because ilis done via the characterization theorem. 26. An affirmative answer is obtained to the first question by combining the results of G. Gratzer, C. R. Platt, and B. Sands [1] with C. R. Platt [lj: every finite transferable lattice is sharply transferable. It is still unknown whether this holds for finite semilattices. 35. B.. M. Schein points out in his review of General Lattice Theory in the Zentralblatt that the problem raised fa characterization of the congruence lattice of a has a satisfactory solution in G. l. Zitomirskii [ll 36. Let L be a semilattice with in which au intervals [0, are pseudocomplemented. Then S(a), the set of pseudocomplements in [0, a], form a boolean aif!lebra. This problem asks for a characterization of the family as a ranges over L. Apparently, the problem admits more than one In a letter of May 1978, B. M. Schein reported that O. M. Mamedov solved this problem using two interpretations while there is no known solution using a third interpretation which is due to Schein. 40. M. E. Adams and J. Sichler [1] proved that in every nontrivial equation class K of lattices, every lattice L in K is the Frattini sublattice of a suitable lattice A in K. Chapter U. The problem of generalization of the uniqueness of an irredundant representation of an element of a distributive lattice (mentioned in the third paragraph of the further Topics and References) has been solved in V. A. Gorbunov [2]. 7. In a recent manuscript, E. T. Schmidt this problem is brought one step closer to solution: every distributive algebraic lattice in which the meet of any two COJrnpact elements is again is isomorphic to the congruence lattice of a lattice. 18 and 19. Solved in the affirmative A. Urquhart (1]: a finite K and a finite group G, there exist a finite L whose congruence lattice is isomorphic to the congruence lattice of K and whose automorphism group is isomorphic to 20. W. A. Lampe showed that there is no fixed type of algebras all algebraic lattices as congruence lattices. The following result is proved in R. Freese,W. A.Lampe,andW.Taylor [1]: LetL bethecongruencelatticeofan infinite dimensional vectorspaceover a field of cardinality m. Then L is notisomorphic to the congruence lattice of an algebra with less than m operations. 22. Ph. Olin proves that there exists an infinite distributive lattice A such that

3 398 G. GRATZER ALGEBRA UNIV. for every distributive lattice B. the distributive free product of A and B is an elementary extension of A. This implies an answer in the negative for this problem. 23. In a letter dated Dec O. M. Mamedov claimed that he had proved that for an algebraic and dually algebraic distributive lattice equational compactness is equivalent to completeness and the two infinite distributive identities. The original problem, namely, whether this equivalence holds for any distributive lattice. is still open. 31. The lattice of quasivarieties of distributive lattices with pseudocomplementation is investigated in V. A. Gorbunov [I] and G. Gratzer, H. Lakser. and R. W. Quackenbush [1]. In the latter it is proved that the principal ideal generated by B 3 has 2 No members and it is nondistributive. Chapter IV. In a remarkable series of papers. R. Freese succeeded in applying the techniques J. von Neumann developed to coordinatize complemented modular lattices to attack some deep lying problems of modular lattice theory. His results include the following: I. For n ~ 4. the lattice of subspaces of an n-dimensional vectorspace over a prime field is a projective modular lattice (R. Freese (I]). 2. The variety of modular lattices is not generated by its finite members (R. Freese [2]). 3. The word problem for the free modular lattice on five generators is unsolvable (R. Freese [3]). Thus we have a negative solution to Problem 5 and to the second question of Problem R. (The first question. whether the word problem for F M (4) is decidable. is still open.) 5. See the discussion above. 6. Not much has been learned about the finite sublattices of free modular lattices. but see B. Jonsson (2] for a discussion of related problems. 8. See the discussion above. 26. S. V. Polin [I] found an equational class K of algebras with the property that a nontrivial lattice identity holds in au congruence lattices of members of K, but K is not congruence modular. There is now a large body of literature dealing with this and related problems. For a review of this new field. see Appendix 3 of B. Jonsson in G. Gratzer [I]. 27. The problem is whether a congruence identity that can be expressed as a Mal'cev type condition implies congruence modularity. Some of the identities discussed in A. Day and R. Freese [I] are likely candidates for the solution of this problem. 28. It is proved in R. Freese and B. Jonsson [1] that modularity and the

4 Vol 11, 1980 General Lattice Theory: 1979 Problem Update 399 arguesian identity are equivalent for congruence lattices of all members of a 31. The problem is to find an effective alogrithm that turns configurational conditions for geometries into equivalent identities for the subspace lattices. The problem should be understood with the concluding phrase "whenever possible". R. Freese pointed out that, for instance, the pappian law is not equivalent to a lattice identity. (However, A. Day [3) has an identity equivalent to the pappian law for projective planes.) The problemshould be rephrased as follows: Find an effective algorithm that determines whether a configurational condition for a geometry is equivalent to an identity for the subspace lattice and if it then find such identities. 32. P. Pudhik and J. Tuma achieved the long awaited breakthrough: every finite lattice is embeddable in a finite partition lattice. 35. A negative solution was obtained in E. R. Canfield [lj: There is a finite partition lattice which contains an antichain that is larger than the longest antichain of elements of the same height. 36. The technique of P. Pudhik and J. Tuma [1] yield a large class of finite lattices that can be represented as congruence lattices of finite algebras. (Curiously, this class is closely related to finite sublattices of a free lattice see A. Day [2] - and thus also to tranferablity.) In a related paper, P. Pudhik and P. P. PaIfy [1] prove that a finite lattice is isomorphic to the congruence lattice of a finite iff it is isomorphic to an interval of the subgroup lattice of a suitable finite group. Chapter v. 9. J. Berman [1] constructs 2 xo equational classes of lattices with the property that the two element chain is in the amalgamation class. However, none of these classes are modular, so the problem is still open. 14. The problem is whether in the lattice of equational classes of lattices every proper interval contains an atom. An affirmative solution is in a footnote to this problem in General Lattice Theory. The footnote was in error, the problem is stib open. 17. Another long standing problem of lattice theory has been solved: the covers of Ns are those listed in Exercise 2.1. This was proved in B. Jonsson and I. Rival [ll 18. It is still an open problem whether there is an eqnational class of lattices with uncountably many covers. The solution reported in a footnote in General Lattice did not work out.

5 400 G. GRATZER ALGEBRA UNIV. 20. A very elegant finite equational basis form v N 5 was exhibited by B. Jonsson [t], providing a.solution to the problem. Chapter VI. I. Let A be C 2 x C:1 and let B be Ct. Then the free product of A and B has a minimal generating set not included in AU B bym. E. Adams [I], providing a negative solution to the problem. For a recent related result see G. Gratzer and A. P. Huhn [I]. 2. The answer to the first question is in the affirmative: the Common Refinement Property holds for bounded free products. In fact. by G. Gratzer and A. P. Huhn [I], it holds for amalgamated free products whenever we amalgamate over a lattice satisfying the Ascending or the Descending Chain Condition. The second question of this problem. whether the analogous result holds in any equational class of lattices. is still open. 26. Every finite monoid is the endomorphism monoid of a finite lattice. This is proved in M. E. Adams and J. Sichler [2], along with a number of results showing the effect of the height of a lattice on its endomorphism monoid. 30 and 31. M. E. Adams and J. Sichler [4] found 2 Xll equational classes of lattices in which every lattice is embeddable in a uniquely complemented lattice (in fact. Theorem VI.3.7 holds). 32. In a series of papers, D. Kelly and the author show that many results of lattice theory can be extended to the m-complete case (see G. Gratzer and D. Kelly [1], [2], and [3] andg. Gratzer, A. Hajnal, and D. Kelly [1]). In particular there are m-complete uniquely complemented lattices. The second part of the problem, whether similar results can be proved in the presence of,join and meet continuity. is still open. 33. The following structure theorem was proved in G. Gratzer. A. P. Huhn, and H. Lakser [1]: Let L be a finitely presented lattice. Then there is a congruence relation e on L such that LIe is finite and every congruence class is embeddable in a free lattice. 35. R. Freese and J. B. Nation [I] show that the automorphism group of a finitely presented lattice is finite. See also G. Gratzer and A. P. Huhn [2]. 45. Theorem VI.4.5 (claiming the existence of two hopfian lattices whose free product is not hopfian) was verified in any equational class of lattices in M. E. Adams and J. Sichler [3]. 49. An affirmative answer follows from W. Poguntke [I]. In that paper the following result is proved: if L is a finitely generated lattice with more than one element satisfying (SD,,), then L has an atom. Consequently, such an L has a prime ideal But it is well known (see, e.g.. G. Gratzer [1]. Theorem 22.6) that if

6 Vol 11, 1980 General Lattice Theory: 1979 Problem Update 401 L is an arbitrary lattice and every finitely generated sublattice of L has a prime ideal, then L itself has a prime ideal. Hence every lattice with more than one element satisfying a semidistributive law has a prime ideal. (This is stronger than the result conjectured in the problem: in the problem it was assumed that the lattice satisfies both semidistributive laws and also Whitman's condition (W).) REFERENCES M. E. ADAMS [I] Generators offree products of lattices. Algebra Universalis 7 (11)77), M. E. ADAMS and J. SICHLER [I] Frattini sublattices in varieties of lattices. Colloq. Math. [2] Bounded endomorphisms of lattices of finite height. Canad. J. Math. 29 (11)77), [3] V-free product of hopfian lattices. Math. Z. 151 (11)76),251)-262. [4] Lattices with unique complementation. Pacific J. Math. J. BERMAN [I] Interval lattices and the amalgamation property. Algebra Universalis. E. R. CANFIELD [I] On a problem of Rota. Advances in Mathematics 24 (l1)7r), A. DAY [I] Splitting lattices generate all lattices. Algebra Universalis 7 (11)77), [2] Characterizations of finite lattices that are bounded homomorphic images or sublattices of free lattices. Canad. J. Math. 31 (11)71)), (1)-7R. [3] In search of a pappian identity. Canadian Math. Bull. A. DAY and FREESE [1] A characterization of identities implying congruence modularity. Canadian J. Math. R. FREESE [I] Projective geometries as projective modular lattices. Trans. Amer. Math. Soc. 251 (11)71)). 321)-342. [2] The variety of modular lattices is not generated by its finite members. Trans. Amer. Math. Soc. 225 (11)71)), [3] Free modular lattices. Trans. Amer. Math. Soc. R. FREESE and B. JONSSON [1] Congruence modularity implies the Arguesian property. Algebra Universalis 6 (1976), R. FREESE, W. A. LAMPE, and W. TAYLOR [1] Congruence lattices of algebras of fixed similarity type. I. Pacific J. Math. 82 (11)71)), 51)-68. R. FREESE and J. B. NATION [1] Finitely presented lattices. Proc. Amer. Math. Soc. 77 (11)71)), V. A. GORBUNOV [I] On lattices of quasivarieties. (Russian.) Algebra i Logika 15 (11)76), [2] Canonical representations in complete lattices. (Russian.) Algebra i Logika 17 (11)78), 41) G. GRATZER [1] Universal Algebra. Second Expanded Edition, Springer Verlag, New York, Berlin, Heidelberg, G. GRATZER, A. HAJNAL. and D. KELLY [1] Chain conditions in free products of lattices with infinitary operations. Pacific J. Math. 83 (1979), G. GRATZER and A. P. HUHN LI] Common refinements of amalgamated free products of lattices. Notices Amer. Math. Soc. 26 (11)71)), A-425.

7 402 G. GRATZER ALGEBRA UNIV. [2] A note on finitely presented lattices. Manuscript. G. GRATZER, A. P. HOON, and H. LAKSER [1] On the structure of finitely presented lattices. Canad. J. Math. G. GRATZER and D. KELLY [1 J Free m-products of lattices. I-III Colloq. Math. [2] When is the free product of lattices complete? Proc. Amer. Math. Soc. 66 (1877). 6-K [3] A normal form theorem for lattices completely generated by a subset. Proc. Amer. Math. Soc. 67 (1 ()77), K G. GRATZER, H. LAKSER. and R W. QUACKENBUSH [1 J On the lattice of quasivarieties of distributive lattices with pseudocomplementation. Acta Sci. Math. Szeged. G. GRATZER and C. R PLATT [IJ A characterization of sharply transferable lattices. Canad. J. Math. 32 (198(), G. GRATZER. C. R PLATT, and B. SANDS [1] Embedding lattices into ideal lattices. Pacific J. Math. 85 (1979), B. JONSSON [1] The variety covering the variety of all modular lattices. Math. Scand. 41 (1977), [2] Varieties of lattices: Some open problems. Algebra Universalis, 10 (1980), B. JONSSON and I. RIVAL [I] Lattice varieties covering the smallest non-modular lattice variety. Pacific J. Math. 82 (l()79), K D. KELLY [I] On the dimension of partially ordered sets. Discrete Math. Ph.OUN [1] Elementary properties of distributive lattice free products. C. R PLAIT [lj Finite transferable lattices are sharply transferable. Proc. Amer. Math. Soc. W. POOUNTKE [1] A note on semidistributive lattices. Algebra Universalis 9 (1979), S. V. POLIN [I] On identities in congruence lattices of universal algebras. (Russian.) Mat. Zametki 22 (1977), P. PUDlAK and P. P. PALFY [1 J Congruence lattices of finite algebras and intervals in subgroup lattices of finite groups. Algebra Universalis 11 (1980). P. PuDLAK and J. TUMA [1] Every finite lattice can be embedded in the lattice of all equivalences over a finite set. Algebra Universalis 10 (1980), E. T. SCHMIDT [1] The ideal lattice ofa distributive lattice with () is the congruence lattice of a lattice. Manuscript. A. URQUHART [I] A topological representation theory for lattices. Algebra Universalis 8 (1978), G. 1. ZITOMIRSKII [1] On the lattice of all congruence relations of a semi-lattice. Ordered sets and lattices, No.1 (Russian.), Izdat. Saratov. Univ., Saratov, University of Manitoba Winnipeg, Manitoba Canada Note added in proof (Sept. 10, 1980): G. Richter has shown that semimodularity and M-symmetry are not equivalent, solving IV.16. R McKenzie and D. Monk have some unpublished correspondence concerning n.ll.