Modular and Distributive Lattices

Transcription

1 CHAPTER 4 Modular and Distributive Lattices Background R. P. DILWORTH Imbedding problems and the gluing construction. One of the most powerful tools in the study of modular lattices is the notion of the projectivity of intervals. The intervals [a, avb] and [at\b, b) are tmnsposes of one another. Two intervals which can be transformed into each other by a series of transposes are said to be projective. For example, in the lattice of Figure 1 every covering interval is projective to the interval [a, e) and hence every two covering intervals are projective. If an interval is collapsed under a homomorphism, then every transpose is also collapsed and hence every interval projective to it is collapsed. Thus a homomorphism of the lattice in Figure 1 is either an isomorphism or it maps the entire lattice onto a one-element lattice. A lattice with this property is said to be simple. In particular, if a simple lattice is imbedded in a direct product oflattices, then it can also be imbedded in one of the component lattices since not all of the homomorphisms onto the components can be trivial. During my graduate years at Caltech I had noticed that two lattices could be combined to form a larger lattice if an upper interval of one lattice was isomorphic to a lower interval of the other lattice. The process was to patch the two lattices together so that corresponding elements under the isomorphism were identified as indicated in Figure 2. This process of patching (now called 'gluing') had the property of preserving modularity and also distributivity. Also if the isomorphic intervals were non-trivial and the lattices were simple, then the resulting lattice would likewise be simple. While I was at Yale, it occurred to me that the patching technique might be relevant to one of the then outstanding problems concerning modular lattices, 205

2 Figure 1 Figure 2 namely, can every modular lattice be imbedded in a complemented modular lattice. It seemed unlikely that there would be an affirmative answer since simple complemented modular lattices were projective geometries which were highly uniform in their structure. I discussed this problem with Marshall Hall who was at Yale at this time and mentioned the patching technique. Almost immediately he came up with the idea of patching the lattice of Figure 1 onto the lattice of subspaces of a non-desarguesian plane to give the lattice schematically represented in Figure 3. If this modular lattice could be imbedded isometrically in a complemented modular lattice L, then, since L is a direct product of projective geometries, the lattice could be imbedded in one of the components which would have to be of projective dimension at least 3. But every plane in a projective geometry of projective dimension at least 3 is Desarguesian. Thus such an imbedding is not possible. Hall went on to show that it could not be imbedded in any way in a complemented modular lattice. We then constructed several other examples of modular lattices which could not be isometrically imbedded in a complemented modular lattice. Meet and join irreducible elements in modular lattices. Meet and join irreducibles have played an important role in many aspects of lattice theory. An element q is meet irreducible if q = x 1\ y implies q = x or q = y. Join irreducibles are defined dually. If a finite dimensional lattice is distributive, there is a natural one-to-one correspondence between the meet and join irreducibles, namely if q is a 206 THE DILWORTH THEOREMS

3 Figure 3 meet irreducible, there is a unique minimal join irreducible p such that p i q. In turn, q is the unique maximal meet irreducible not containing p. It follows from this correspondence that a finite distributive lattice has the same number of meet and join irreducibles. Although there is no natural correspondence between meet and join irreducibles in a finite modular lattice it was observed in the early 1940's that the number of meet irreducibles was the same as the number of join irreducibles. This was stated as a conjecture by Schiitzenberger but for a number of years no proof was forthcoming. In the early 1950's I decided to make a major attack on the problem. Since meet irreducibles are those elements covered by a single element of the lattice and join irreducibles are those elements covering a single element of the lattice, it was natural to attack the more general problem of showing that IVkl = IWkl where Vk is the set of elements of the lattice covered by precisely k lattice elements, Wk is the set of elements covering precisely k elements, and lsi denotes the number of elements in the set S. This equality had been known for some time in the case of the lattice of subspaces of a projective geometry. Since a finite complemented lattice is a direct product of projective geometries, it follows that the equality holds for all finite complemented modular lattices. I spent some time trying to describe the lattice structure of a finite modular lattice in terms of its complemented sublattices. However, this seemed to be a more complicated problem than the original combinatorial problem. It did become clear from these early investigations that the complemented modular sublattice M generated by the atoms of the lattice L would playa significant role. This led to the idea of reducing the problem from the lattice L to the sublattices L m, where me M and Lm consists of the elements of L containing m. If it turned out to be possible to express IVk I in terms of the corresponding numbers for lattices Lm and the same expression also works for IWkl, then the result would follow by induction on the dimension of the lattice. Now there was available a standard device for counting elements in an ordered set, namely, the Mobius function. For m E M, f..l(m) is Modular and Distributive Lattices 207

4 defined by the recurrence relation x$m ~ J l ( x ) = { ~ if m equals z, the null element of the lattice, if m does not equal z. From this basic property of the Mobius function it was quite direct to show that IVkl was expressible as a linear combination of the corresponding numbers for the lattices Lm with Mobius coefficients. With some additional work and making use of the fact that equality holds in the complemented case, I was able to show that the same linear relation held for IWkl in terms of the corresponding number of the lattices Lm. The theorem then follows by induction. Distributivity in lattices. A closure operation on a set S is a mapping c.p from subsets of S to subsets of S with the following properties: (1) T ~ c.p(t) for all T ~ S; (2) c.p(u) ~ c.p(t) if U ~ T; (3) c.p(c.p(t)) = c.p(t) i f T ~ S ; If S is an ordered set, a closure operation c.p is called an imbedding operator if ( 4 ) c.p( { s }) = (s) for all s E S, where (s) denotes the set of all xes such that x ~ s. A subset T of S is said to be c.p-closed if c.p(t) = T. The c.p-closed subsets form a complete lattice under the operation of set intersection and the closure of the set union. If S is an ordered set, the mapping s 1-+ (s) imbeds S in the lattice of c.p-closed subsets of S. Imbedding operators have a natural ordering given by c.p ~ 'IjJ if and only if c.p(t) ~ 'IjJ(T) for all T ~ S. The minimum operator w has the defining property that w(t) consists of all x such that x ~ t for some t E T. The maximum operator v is defined by letting v(t) be the set of all lower bounds of the upper bounds of T. The starting point of the paper "Distributivity in Lattices" was the observation that the usual distributivity of meet with respect to join in a lattice L can be expressed in terms of an imbedding operator fj as a /\ fj(t) = (J( a /\ T) where a /\ T = {a /\ t : t E T} and fj(t) is the ideal of L generated by T, i.e., the set of all elements of L contained in a finite join of elements of T. ReplaCing (J by an arbitrary imbedding operator c.p produces many possibilities for distributivity conditions in lattices. This raised a number of questions concerning the relationships between the different distributivities and the relation to the lattice structure of the imbedding operators. I proposed to investigate these questions and was joined in the study by an innovative young graduate student, Jack McLaughlin. It is a natural conjecture that if c.p 2: 'IjJ then c.p-distributivity should imply 'IjJ-distributivity. However, this turns out not to be the case. In fact, there exists an imbedding operator c.p on the Boolean algebra of order 8 such that the Boolean 208 THE DILWORTH THEOREMS

5 algebra is not <p-distributive, while it is also true that every Boolean algebra is v distributive. <p-distributivity is related to v-distributivity in the following way: L is <p-distributive if and only if the lattice of <p-closed subsets of L is v-distributive. But for complete lattices v-distributivity is equivalent to ordinary infinite distributivity. Hence, <p-distributivity of L is equivalent to infinite distributivity of the lattice of <p-closed subsets of L. Given an imbedding operator <p, suitable collections of subsets of S provide associated imbedding operators contained in <po These associated operators have consistency properties for distributivity not found among imbedding operators in general. It turns out that all of the common imbedding operators are associated with the normal imbedding operator v. Editors' note. The paper "Aspects of Distributivity," published in 1984, is in part a survey of distributivity and gives much insight into Dilworth's thinking about this area of lattice theory. In addition, Dilworth proved that in a distributive lattice, an irredundant decomposition into irreducibles is unique (if it exists), even when the ascending chain condition fails. Modular and Distributive Lattices 209