May 1, 1996 - shall refer to nite distributive lattices with this property as being ( nitely) representable. Recall that the category of nite distributive (0;1)-lattices ...
Maximal sublattices of nite distributive lattices M. E. Adams, Ph. Dwinger and J. Schmid May 1, 1996
Abstract Algebraic properties of lattices of quotients of nite posets are considered. Using the known duality between the category of all nite posets together with all order-preserving maps and the category of all nite distributive (0 1)-lattices together with all (0 1)-lattice homomorphisms, algebraic and arithmetic properties of maximal proper sublattices and, in particular, Frattini sublattices of nite distributive (0 1)-lattices are thereby obtained. ;
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Addresses: Dept. of Math. and Comp. Sci., SUNY, New Paltz, NY 12561, USA; Dept. of Math., Stat. and Comp. Sci., Univ. of Illinois, Chicago, IL 60607, USA; Math. Institut, Univ. of Bern, CH-3012 Bern, Switzerland
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1 Introduction The intersection of all maximal proper sublattices of a distributive lattice L is called the Frattini sublattice of L, denoted (L). Even though it is known that, for each distributive lattice L, there exists a distributive lattice K such that L = (K ) (see [2]), in general jK j jLj + @ . It is always possible to nd K for which jK j = jLj + @ . However, there exist nite distributive lattices L such that K is by necessity in nite (see [2] and [6]). Recently, in [1], the class of nite distributive lattices L for which there exists a nite distributive lattice K with L = (K ) were characterized. We shall refer to nite distributive lattices with this property as being ( nitely) representable. Recall that the category of nite distributive (0; 1)-lattices with (0; 1)lattice homomorphisms is dually equivalent to the category of nite partially ordered sets with order-preserving maps. Under this duality, a (necessarily nite) poset is said to be representable if and only if the associated distributive (0; 1)-lattice is. In fact, representable posets were characterized in [1]. As shown there, determining whether a nite poset is representable is decidable, however, as established in [14], doing so is NP-complete. In the present paper, lattices of quotients (see De nition 3.1) of nite posets are investigated. Since one-to-one (0; 1)-lattice homomorphisms correspond to onto order-preserving mappings, the lattice of quotients of a nite poset is dually isomorphic to the lattice of subalgebras of the corresponding distributive (0; 1)-lattice. Under this isomorphism, the atoms of the lattice of quotients correspond to the maximal proper sublattices of the distributive (0; 1)-lattice. In Section 3, algebraic properties of the atoms of the lattice of quotients are considered. Since the notions of vertical and horizontal contractions (see De nition 3.3) are central to the characterization of representable posets as given in [1], of particular interest in this section are the results directly related to these concepts (see Propositions 3.6, 3.10, and 3.11). Although applications of the results of Section 3 are primarily reserved for Sections 4 and 5, an immediate consequence contained in this section is the derivation of the characterization given in [1], albeit in a di erent guise (see Corollary 3.12). It is well known that epimorphisms in the variety of distributive (0; 1)0
0
2
lattices need not be onto (see Section 2). This leads naturally to the notions of epic sublattice (see De nition 2.1) and epic Frattini sublattice e (L) (the intersection of all proper maximal epic (0; 1)-sublattices of a nite distributive (0; 1)-lattice L). Our main results are to be found in Section 4, where a greater understanding of the lattice theoretic r^ole played by vertical and horizontal contractions in the determination of the Frattini sublattice is obtained. For example, it is shown that (L) = e(L) if and only if the poset associated with L is vertically contracted (see Theorem 4.2). Also, it is shown that (L) is an epic sublattice of L if and only if the associated poset is both vertically and horizontally contracted (see Theorem 4.3). Perhaps surprisingly, in addition it is shown that if K is the distributive (0; 1)-lattice obtained by vertically contracting the poset associated with L, then K is the largest sublattice of L such that (L) = e(K ) (see Theorem 4.4). Moreover, if K is obtained by vertically contracting and then horizontally contracting the poset associated with L (cf., the characterization of representable posets as given by Corollary 3.12), then K is the largest sublattice of L such that (L) = e (K ) and (L) is an epic sublattice of K . The section concludes by considering how, in the present context, a lattice may be represented. For example, it is shown that L = e(K ) for some K if and only if L is representable (see Theorem 4.5). In Section 5, the results of Section 3 are applied to obtain arithmetic properties of nite distributive lattices concerning both the number (see Propositions 5.1 and 5.2) and size (see Proposition 5.5) of their maximal proper sublattices.
2 Preliminaries and Notation We write D01 for the category of all nite distributive lattices having universal bounds 0 6= 1 with lattice homomorphisms preservig these bounds, and P for the category of all nite nonvoid partially ordered sets (also called posets) with order-preserving maps. For L 2 D01, J (L) denotes the poset of all non-zero join-irreducibles of L under the order inherited from L; similarly, M(L) stands for the poset of all non-unit meet-irreducibles of L. We use \join-reducible", \meet-reducible", \doubly reducible" for elements outside 3
J (L), M(L), J (L) [ M(L), respectively, on the other hand, \doubly irreducible" for those in J (L) \ M(L). For P 2 P, O(P ) denotes the lattice of
all order ideals (or down-sets or decreasing sets) in P with set operations of join and meet. Note that ; 2 O(P ). We write pkq to indicate that p and q are incomporable elements of P . We also speak of join-irreducible elements of a ( nite) poset, meaning those having exactly one lower cover, and dually of meet-irreducibles. Finally, for any set S , jS j denotes its cardinality. J and O as de ned set up the object part of the so-called Birkho duality between the categories D01 and P. This duality is, in fact, a category coequivalence, and we will also use J and O for the (contravariant) functors involved. Recall ([4], IV.2) that for h 2 HomD01 (L ; L ) one has (i) h is one-to-one if and only if h is monic if and only if J (h) is onto, and (ii) h is epic if and only if J (h) is one-to-one if and only if J (h) is monic. It follows that monomorphisms in D01 as well as in P are the same as one-toone morphisms:; moreover, epimorphisms and onto morphisms coincide in P (indeed, if f 2 P is epic, then O(f ) 2 D01 is monic and thus J (O(f )) = f is onto). However, epimorphisms in D01 need not be onto, see [4], p. 110 for the standard example. We turn our attention to sublattices. Working in D01 means that all sublattices contain the 0 and 1 of their parent lattice. Sublattices correspond to onto morphisms in P; this fact provides the base for the material in Section 3. A special type of sublattice is singled out by 1
2
De nition 2.1 A sublattice K of L 2 D01 is called an epic sublattice if the canonical inclusion map is an epimorphism.
The de nition applies to any equational category of (universal) algebras; see [4], I.20, De nition 12. Epic sublattices are also characterized by the following
Proposition 2.2 Let L; K 2 D01, K a sublattice of L. Then K is an epic sublattice of L if and only if jJ (K )j = jJ (L)j. Proof: Let i : K ?! L be the canonical inclusion. Then i is epic (and one-to-one) if and only if J (i) is one-to-one (and onto). Hence, K is an epic 4
sublattice of L if and only if J (i) : J (L) ?! J (K ) is a bijection.
2
The following de nition is standard in its rst part:
De nition 2.3 Let L 2 D01. Then (i) the Frattini (sub-)lattice (L) of L is the intersection of all proper
maximal (0; 1)-sublattices of L, and (ii) the epic Frattini (sub-)lattice e(L) of L is the intersection of all proper maximal epic (0; 1)-sublattices of L.
Since the intersection of an empty family of subsets of some set X is X itself, the de nition takes care of the degenerate cases arising for L = 2, the 2-element chain, and L = 3, the 3-element chain: (2) = e(2) = 2, while (3) = 2 and e(3) = 2. Again, the de nition applies to any equational category of (universal) algebras. For (L), there is an alternative description: Call x 2 L a nongenerator if G n fxg still generates L whenever G L is a generating set for L, and write N (L) for the set of all nongenerators of L. Then we have
Fact 2.4 (L) = N (L). Proof: Consider an arbitrary proper maximal sublattice M of L and x 2 L n M . Then M [fxg generates L while M does not, so any element outside (L) fails to be a nongenerator. Conversely, assume x 2 (L) and x 2 G for some generating set G L. If G n fxg does not generate L, there exists some proper maximal sublattice M of L containing G n fxg, contradicting x 2 (L). 2
We recommend the references given in [1] and [14] as a point of entry to the literature on Frattini lattices (also for non nite and non distributive lattices). Working in D01 here allows the use of Birkho duality to the fullest extent possible. We are aware of the fact that a large part of the literature considers { in the nite, distributive case { rather the category 5
D n of nite distributive lattices with arbitrary lattice homorphisms (not necessarily preserving universal bounds). Choosing D01 instead is however not a serious restriction, as the following considerations are intended to show: Let L 2 D01. By a general sublattice of L we mean any subset of L closed under join and meet, and the general Frattini (sub-)lattice of L, denoted by 0 (L), is the intersection of all proper maximal general sublattices of L. The relation between (L) and 0(L) is given by
Fact 2.5 0 (L) = (L) n (f0L; 1Lg \ (J (L) [ M(L))). Proof: If 0L 2= M(L) and 1L 2= J (L), general maximal proper sublattices of L are the same as proper maximal sublattices (see [11]). If 0L 2= M(L) but 1L 2 J (L), the only general maximal proper sublattice of L which is not a (0; 1)-sublattice is L n f1Lg. The remaining two cases are similar. 2
3 Quotients, critical pairs, contractions We consider an arbitrary but xed poset P 2 P in this section.
De nition 3.1 A quotient of P is a pair (f ; P ) where P is a poset and f : P ?! P an order-preserving map from P onto P . The set of all quotients of P is denoted by Q (P ). Q (P ) is partially ordered by (f ; P ) 1
1
1
1
1
1
1
1
(f ; P ) if and only if there exists an order-preserving (necessarily onto) map f : P ?! P such that ff = f (where (f ; P ) and (f ; P ) are identi ed if f is an order-isomorphism). 2
2
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2
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Under the order just de ned, Q (P ) is a lattice which is dually isomorphic to the lattice of sublattices of L := O(P ) by Birkho duality (see section 2). Moreover, if (f ; P ) 2 Q (P ), then Q (P ) is isomorphic with the interval [(f ; P )) of Q (P ), the required isomorphism being given by (f ; P ) 7?! (f f ; P ) for (f ; P ) 2 Q (P ). 1
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2 1
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The set of all atoms of Q (P ) will be denoted by AtQQ(P ). Obviously, these atoms correspond to the maximal proper sublattices of L, W and AtQQ(P ) corresponds to (L). Extending the notation, we write W AtQQ(P ) =: ('; (P )).
De nition 3.2 Let p; q 2 P , p 6 q. The pair (p; q) is called critical i for any a; b 2 P , a > p implies a q, and b < q implies b p. We write M (P ) { or simply M , if P is clear from the context { for the set of all such pairs.
We note that critical pairs have been considered under various names, in di erent degrees of generalization, and in many places: see, for example, [7], [8], [10] and [12]. We point out that the term \critical pair" usually is reserved for the members of B (P ) M (P ) as given below. We follow the usage as in [14] and allow p q in a critical pair (p; q) (which still excludes p = q by 3.2). The following subsets of M (P ) will be particularly important for our purposes: A(P ) := f(p; q) 2 M (P ) : p < qg B (P ) := f(p; q) 2 M (P ) : pkqg C (P ) := f(p; q) 2 B (P ) : (q; p) 2 B (P )g B 0 (P ) := B (P ) n C (P ) If P is clear from the context, we write A, B , C and B 0 for A(P ) etc. The following observations will be useful:
(p; q) 2 A if and only if p is a lower cover of q, p is meet-irreducible
and q is join-irreducible. For (p; q) 2 B , the stronger implications \if a > p then a > q" and \if b < q then b < p" hold.
Hashimoto [9] was the rst to observe that there is a bijective correpondence between the critical pairs of P on one side and the atoms of Q (P ) on the other side (and thus with the proper maximal sublattices of O(P )). For (p; q) 2 M , we de ne (tpq ; Ppq ) as follows: 7
If (p; q) 2 A, let Ppq := P n fqg with the order induced from P , and de ne tpq : P ?! Ppq by tpq (a) := a for a 6= q, tpq (q) := p. Note that tpq (a) < tpq (b) implies a < b for any a; b 2 P . If (p; q) 2 B , let Ppq be the poset with underlying set P and partial order [(q; p) where is the order of P , and de ne tpq to be the identity map.
Accordingly, we will write AtAQ (P ) := f(tpq ; Ppq ) : (p; q) 2 A(P )g and speak of the A-atoms of P ; similarly AtB Q (P ) etc. As already apparent in [1] and [14], the decisive critical pairs are those in B 0(P ). This is additionally substantiated by the following simple observation which exhibits the close connection between A-pairs and C -pairs: given (p; q) 2 A(P ), let P ? be the poset with carrier P and order P nf(p; q)g. It is easy to check that (p; q) 2 C (P ?). Conversely, assume (p; q) 2 C (P ) and de ne P to be the poset with carrier P and order P [f(p; q)g; then again by a straightforward check we have (p; q) 2 A(P ). Of course P = P? in the rst and P = P ? in the second case. The following de nition is intended to capture this fact. +
+
+
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De nition 3.3 ([1]) P is called vertically contracted (abbreviated v.c.) if and only if A(P ) = ;, and horizontally contracted (abbreviated h.c.) if and only if C (P ) = ;. Let v be the least equivalence relation containing A(P ). v has the following properties which are easy to check (see, for example, [1]): writing [a]v for the equivalence class of a 2 P , we have that [a]v is a chain in P . Accordingly, for a 2 P we put a := max[a]v and a := min[a]v . If [a]v 6= [b]v , then for any x 2 [a]v , y 2 [b]v , we have x < y i a < b. We write V (P ) for the quotient set P= v and partially order V (P ) by [a]v < [b]v i a 6v b and a < b. Let v : P ?! V (P ) be the canonical map; v is order-preserving and v(a) < v(b) implies a < b. Finally, A(V (P )) = ; and thus V (P ) is v.c..
Remark 3.4 It follows from the above properties that nontrivial v -classes
are chains [a; b] with a meet-irreducible but join-reducible, b join-irreducible 8
but meet-reducible and x doubly irreducible for a < x < b. This condition is the dual of the one characterizing maximal sublattices of nite distributive lattices in [5, Theorem 4] or in [13, Theorem 3]: A sublattice S is maximal if and only if L n S is either trivial (consisting of a single doubly irreducible element) or of the form [a; b] with a join-irreducible but meet-reducible, b meet-irreducible but join-reducible and x doubly reducible for a < x < b. Next, let h be the least equivalence relation containing C (P ). It is easy to check that h = (P ) [ C (P ), indeed, a h b if and only if a and b share the same sets of upper respectively lower covers. Writing [a]h for the h-class of a 2 P , it follows that [a]h is an antichain in P ; and if [a]h 6= [b]h , then for any x 2 [a]h, y 2 [b]h we have x < y i a < b. We write H (P ) for the quotient set P= h and partially order H (P ) by [a]h < [b]h i a 6h b and a < b. Let h : P ?! H (P ) be the canonical map; g is order-preserving and g(a) < g(b) if and only if a < b. Finally, C (H (P )) = ; and thus H (P ) is h.c.. Again, details may be found in [1]. Since v -classes are chains and h-classes antichains, the following is immediate:
Lemma 3.5 v \ h= f(a; a); a 2 P g.
2
Proposition 3.6W In Q (P ), (i) (v; V (P )) = W AtAQ (P ), and (ii) (h; H (P )) = AtC Q (P ).
Proof: For (i), we show rst that (v; V (P )) (tpq ; Ppq ) for (p; q) 2 A. To this end, de ne f : Ppq ?! V (P ) by ftpq = v; f is well-de ned and order-preserving. Next, suppose (g; R) 2 Q (P ) and (g; R) W AtAQ (P ). De ne g : V (P ) ?! R by g v = g. Then g is well-de ned and order-preserving, thus (g; R) (v; V (P )). (ii) Similar. 1
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The following lemma collects the technicalities we will need concerning types of pairs and contractions. 9
Lemma 3.7 (i) Let (p; q) 2 A(P ). Then (a; q) is not critical whenever a= 6 p, and (p; b) is not critical whenever b 6= q. (ii) Let (p; q) 2 B (P ). Then p = p and q = q. (iii) If (p; q) 2 B (P ), then (v(p); v(q)) 2 B (V (P )). Conversely, if (v(p); v(q)) 2 B (V (P )), then (p; q) 2 B (P ). (iv) (p; q) 2 B 0(P ) i (h(p); h(q)) 2 B 0(H (P )). Proof: (i) Since (p; q) 2 A, we have p < q. Assume a 6= p and (a; q) is critical. This implies p a since p < q. As p 6= a, we have p < a. Thus a q since (p; q) is critical. But a 6 q since (a; q) is critical. The other half
of the statement is proved by the dual argument. (ii) Suppose p < p, then p > q. But this implies that q is comparable with p, contradicting (p; q) 2 B . The dual argument takes care of the other half of the claim. (iii) Suppose (p; q) 2 B (P ). Then pkq, and thus v(p)kv(q). Let v(a) > v(p), then a > p; this implies a > q and thus v(a) v(q). Similarly, v(b) < v(q) implies v(b) v(p). Thus (v(p); v(q)) 2 B (V (P )). Conversely, assume (v(p); v(q)) 2 B (V (P )). Then v(p)kv(q), pkq and pkq . Consider a 2 P with a > p. Thus a 2= [a]v and v(a) > v(p) = v(p). Hence v(a) > v(q) since (v(p); v(q)) 2 B (V (P )). This gives a > q and thus a > q. Similarly, b < q implies b < p. Consequently, (p ; q) 2 B (P ). (iv) Assume (p; q) 2 B 0 (P ), then h(p) 6= h(q). Since h(a) < h(b) implies a < b for any a; b 2 P , we must have h(p)kh(q). Suppose h(a) > h(p), then a > p and thus a > q since (p; q) 2 B 0(P ). Consequently, h(a) > h(q). Similarly, h(b) < h(q) implies h(b) < h(p) and so (h(p); h(q)) 2 B (H (P )). If also (h(q); h(p)) 2 B (H (P )), then h(p) h h(q). Since H (P ) is h.c., it follows that h(p) = h(q). This contradiction proves (h(p); h(q)) 2 B 0(H (P )). Conversely, suppose (h(p); h(q)) 2 B 0 (H (P )). Then pkq. Let a > p, then h(a) > h(p) and thus h(a) > h(q) since (h(p); h(q)) 2 B 0 (H (P )), and so a > q. Similarly, b < q implies b < p and (p; q) 2 B (P ). If also (q; p) 2 B (P ), then h(p) = h(q). This contradiction proves (p; q) 2 B 0 (P ).
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Lemma 3.8 Let P be horizontally contracted with partial order . Then W := [f(q; p) : (p; q) 2 B 0(P )g is a partial order on P , and (idP ; (P; )) = AtB Q (P; ). 0
Proof: Writing q 0p for (p; q) 2 B 0(P ), q 0p implies qkp and excludes p 0q. To establish transitivity of , assume q p r and q = 6 p, p =6 r. There 0 are three possible cases; suppose rst that q p < r. Since (p; q) 2 B 0, r > p implies r > q and thus q r. The case q < p 0r is handled in the same way. In the remaining case q 0p 0r, we may suppose q 6 r for otherwise q r trivially. Now observe that a > r implies a > p implies a > q and b < q implies b < p implies b < r for any a; b 2 P since (p; q) and (r; p) are in B 0 (P ). Especially, q > r is excluded, so qkr and (r; q) is critical, that is, (r; q) 2 B (P ). But P is h.c., thus (r; q) 2 B 0 (P ) and so q r as desired. Antisymmetry of is immediate: q 0p 0q is impossible since p is h.c. and q 0p < q and q < p 0q are ruled out by qkp. is re exive trivially and thus an order relation as claimed.
2
By 3.7(iii) (p; q) 2 B (P ) implies (v(p); v(q)) 2 B (V (P )). Accordingly, there exists a map fb : AtB Q (P ) ?! AtB Q (V (P )), given explicitly by fb(tpq ; Ppq ) := (tv p v q v; V (P )v p v q ) for any (p; q) 2 B (P ). 3.7(iv) implies, analogously, the existence of a map fb : AtB Q (P ) ?! AtB Q (H (P )), given explicitly by fb (tpq ; Ppq ) := (th p h q ; H (P )h p h q ) for any (p; q) 2 B 0 (P ). ( ) ( )
( ) ( )
0
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( ) ( )
( ) ( )
Lemma 3.9 (i) fb is onto and one-to-one. Moreover, s fb(s) for any s 2 AtB Q (P ). (ii) fb is onto, and s fb (s) for any s 2 AtB Q (P ). 0
0
0
Proof: (i) fb is onto by 3.7(iii) and one-to-one by 3.7(ii). De ne a map g : Ppq ?! V (P )v p v q by gtpq = tv p v q v. Then g is well-de ned since tpq is the identity map. We show that g is order-preserving, whence s fb(s) by the de nition of order in Q (P ). So let tpq (a) < tpq (b). If (a; b) = 6 (q; p), then a < b in P . Thus v(a) v(b) and tv p v q v(a) tv p v q v(b), hence ( ) ( )
( ) ( )
( ) ( )
11
( ) ( )
gtpq (a) gtpq (b) as desired. If (a; b) = (q; p) then (v(p); v(q)) 2 B (V (P )) by 3.7(iii), and thus tv p v q v(q) < tv p v q v(p) by the de nition of order in V (P )v p v q , and so gtpq (q) < gtpq (p) as desired. (ii) Similar, using 3.7(iv). 2 ( ) ( )
( ) ( )
( ) ( )
Proposition 3.10 W AtQQ(P ) = W AtQQ(V (P )). Proof: We identify Q (V (P )) with [(v; V (P )) Q (P ). Let s = (tpq ; Ppq ) 2 AtB Q (P ). Suppose s (v; V (P ); then for some f : Ppq ?! (v; V (P )), f order-preserving, we have v = ftpq . Since tpq (p) > tpq (q), this implies v(p) v(q) and thus p q, contradicting (p; q) 2 B . Hence s 6 (v; V (P )) and thus s _ (v; V (P )) > (v; V (P )). Lemma 3.9(i) implies fb(s) s _ (v; V (P )) and fb(s) is an upper cover of (v; V (P )). Hence s _ (v; V (P )) = fb (s). Moreover, by 3.9(i) again, every upper cover of (v; v(P )) is obtained in this way (since V (P ) is v.c.,Wthere are noWA-atoms in QW(V (P ))). Putting all this together, we have that AtQQ(P ) = AtAQ (P ) _W AtB Q (P ) = (v; V (P )) _ W W At Q ( P ) = f ( v; V (P )) _ s : s 2 AtB Q (P )g = ffb(s); s 2 AtB Q (P )g = W B W AtB Q (V (P )) = AtQQ(V (P )).
2
Arguing as in the proof of 3.10, but using 3.9(ii), gives the following proposition:
Proposition 3.11 W AtB Q (P ) = W AtB Q (H (P )). Corollary 3.12 ('; (P )) = W AtB Q (H (V (P )))
2
0
Proof: ('; (P )) = W AtQ Q(P ) = W AtQQ(V (P )) (by 3.10)W = W AtB Q (V (P )) W
(since V (P ) is v.c.) = AtB Q (H (V (P ))) (by 3.11) = AtB Q (H (V (P ))) (since H (V (P )) is h.c.). 0
2
Corollary 3.12 is indeed the key result of [1] saying that P is the dual of (L) if and only if P may be obtained by contracting rst vertically, then horizontally the dual of L and nally extending the poset so obtained by adding all its reversed B 0 -pairs to its order relation, utilizing Lemma 3.8. 12
4 The lattice interpretation We consider an arbitrary but xed lattice L 2 D01 with dual poset P := J (L). The purpose of this section is to analyze the signi cance of the operators V and H { de ned in P { in the dual category D01. Extending notation, we de ne V (L) := O(V (J (L))) and H (L) := O(H (J (L))) and thus simply \lift" V and H from P to D01. Using Corollary 3.12, we obtain a chain of sublattices (L) H (V (L)) V (L) L. Recall (Def. 2.3) that the \epic Frattini lattice" e (L) of L is de ned as the intersection of all epic proper maximal sublattices of L. By 2.2 a sublattice of L is epic if and only if the cardinality of its dual equals that of the dual of L { equivalently, if and only if its dual (f; P 0) 2 Q (P ) has f one-to-one. Hence the epic maximal sublattices correspond exactly to the B -atoms of Q (P ) since the map tpq isW bijective if and only if (p; q) 2 B (P ). Consequently, the dual of e (L) is AtB Q (P ) =: ('e; e(P )), again extending notation in an obvious way. By the above, a sublattice of L containing an epic sublattice must itself be epic. So a maximal (among all proper sublattices) and epic sublattice is the same as a sublattice maximal among all proper epic sublattices. So we will speak unambiguously of epic maximal sublattices, mostly dropping \proper" to unburden the language. Since e(J (L)) is the codomain of W AtB Q (J (L)), the following proposition is an easy consequence of Propositions 3.10 and 3.11 and of Corollary 3.12:
Proposition 4.1 (L) = e (V (L)) = e(H (V (L))).
2
The following two theorems highlight the close connection between contracted dual posets on one side and epic sublattices on the other:
THEOREM 4.2 The following are equivalent: (i) (L) is the intersection of epic maximal (proper) sublattices of L. (ii) (L) = e(L). (iii) J (L) is vertically contracted. 13
Proof: (i)!(ii): Since (L) is the intersection of all maximal sublattices, it is then the intersection of all epic maximal sublattices, in particular. (ii)!(i) is trivial. (ii)!(iii): In the dual setting, the hypothesis is W AtQQ(P ) = W AtB Q (P ). Suppose P is not v.c., that is, there is an A-pair (p; q) 2 M (P ). We W show Wthat (tpq ; Ppq ) W6 AtB Q (P ), thus violating the hypothesis. By W
3.11, AtB Q (P ) = AtB Q (H (P )) = AtB Q (H (P )). The codomain of AtB Q (H (P ))Wis (H (P ); ) with the order given by Lemma 3.8. So if (tpq ; Ppq ) AtB Q (P ), we have order-preserving maps f : Ppq ?! ((H (P ); ) (by the de nition of order in Q (P )) and idH P : H (P ) ?! ((H (P ); ) (by 3.8) such that ftpq = idH P h. Since tpq (p) = tpq (q), this implies h(p) = h(q) and thus p h q. But (p; q) 2 A(P ) gives p Wv q and thus p = q by 3.5, contradicting (p; q) 2 A(P ). Hence (tpq ; Ppq ) 6 AtB Q (P ) as desired. (iii)!(ii): By 4.1.
W
0
0
(
(
)
)
2
THEOREM 4.3 The following are equivalent:
(i) (L) is an epic sublattice of L. (ii) e(L) is an epic sublattice of L. (iii) J (L) is vertically and horizontally contracted.
Proof: By 4.2, (iii) implies (L) = e(L). Thus, as (i) also implies (L) = e(L), it is sucient Wto show (i)$(iii). We work again in the dual setting. By 3.12 ('; (P )) = AtB Q (H (V (P ))), or, using 3.8, ' = idH V P hV P v with hV P : V (P ) ?! H (V (P )) the horizontal contraction of V (P ). Now (L) is an epic sublattice of L if and only if ' : P ?! (P ) is bijective, and this is the case if and only if both v and hV P are bijective, that is, if and only if P is both vertically and horizontally contracted. 0
(
(
(
))
(
)
)
(
)
2
While P is v.c. if and only if P = V (P ), H (V (P )) need not be v.c., thus only the implication \P v.c. and h.c. ) P = H (V (P ))" is valid in general. Nevertheless, the sublattices V (L) and H (V (L)) of L have analogous maximality properties: 14
THEOREM 4.4 Let L 2 D01. Then the following hold:
(i) V (L) is the largest sublattice K of L such that (L) = e(K ). (ii) H (V (L)) is the largest K such that (L) = e (K ) and (L) is an epic sublattice of K .
Proof: (i) Assume KW L and (L) = We (K ). Let (g; P 0) 2 Q (WP ) be the
dual of K . We haveW AtB Q (H (P 0)) = AtB Q (P 0) (by 3.11) = AtQQ(P ) (by hypothesis) = AtQQ(V (P )) (by 3.10). Suppose there exists (p; q) 2 A(P ) such that g(p) 6= g(q). It follows that g(p) < g(q), thus hP (g(p)) < hWP (g(q)) since hP -classes are antichains. So the images Wof p and q in AtB Q (H (P 0)) are distinct, while they obviously coincide in AtQQ(V (P )). This contradiction implies g(p) = g(q) for all (p; q) 2 A(P ). De ne f : V (P ) ?! P 0 by f ([a]v ) := g(a); f is well-de ned, order-preserving and surjective. Hence K = O(P 0) is a sublattice of V (L) = O(V (P )) and (i) follows from 4.1. (ii) Assume K L, (L) = e(K ) and (L) is an epic sublattice of K . By part (i), K is a sublattice of V (L). So the dual P 0 of K is an epimorphic image of V (P ) and P 0 has the same cardinality as the dual of (L). The argument proceeds as for part (i): Any pair (p; q) 2 C (V (P )) must be collapsed by the surjection from V (P ) onto P 0 { otherwise, p and q are mapped to di erent points in the dual of (L). This gives a surjection from H (V (P )) onto P 0 and, by 4.1, (ii) follows. Details are left to the reader. 0
0
0
0
0
2
We now turn to the question which members of D01 actually arise as (K ) or e(K ) for some K 2 D01. This question motivated [1]; the answer provided there is essentially the statement of 3.12 readWin the reverse order: L is such if and only if its dual P may be obtained as AtB Q (P 0) for some h.c. poset P 0 living on the same carrier set as P . Posets P with this property are called nitely representable in [1]. Since we will look at three di erent related concepts at the same time, we simplify the language somewhat, as follows: L 2 D01 is called representable (by K ) if and only if L = (K ) for some K 2 D01. It is called e-representable (by K ) (\epically representable") if and only if L = e(K ), and s-e-representable (by K ) (\strongly epically 0
15
representable") if and only if L = e(K ) and L is an epic sublattice of K . As in [1], we extend these de nitions to cover the dualized statements about members of P.
THEOREM 4.5 The following are equivalent for L 2 D01:
(i) L is representable. (ii) L is e-representable. (iii) L is s-e-representable.
Proof: (i)!(ii): If L is representable by K , then L is e-representable by
V (K ) (Theorem 4.4). (ii)!(iii): If L is e-representable by K , then it is s-e-representable by H (K ) (Theorem 4.4). (iii)!(ii) is trivial. It remains to establishW (ii)!(i). We work in the dual setting and assume the dual of L is given as AtB Q (P ) for some (not necessarily v.c.) poset P . We will construct a v.c. posetW P 0 such that H (PW0) = H (P ); then H (P ) = W 0 H (V (P )) and AtB Q (P ) = AtB Q (H (P )) = AtB Q (H (V (P 0))) which shows that L is representable by O(P 0). P 0 is obtained from P by replacing every a 2 P by a 2-element antichain fa0; a00 g and extending the order by giving a0 and a00 the same upper and lower covers as a. The resulting poset P 0 is obviously v.c. and we see that H (P 0) = H (P ). This completes the proof. 2 0
0
Note that P 0 as constructed above serves essentially the same purpose as Q constructed in the proof of Lemma 3.12 in [1].
5 Counting In this section, we will use the results of Section 3 to obtain some arithmetical properties of nite distributive lattices. 16
Fix L 2 D01 and P = J (L) 2 P. We use lower case greek letters to denote natural numbers in this section; in particular for cardinalities like L := jLj and := jP j (or simply and when the context is clear) . We write L (or simply ) for the number of maximal proper (0; 1)?sublattices of L, thus L = jM (P )j. It will turn out to be convenient to consider also general sublattices of L as de ned in Section 2, their number will be written 0L (or simply 0). By the proof of 2.5 we have = 0 if and only if 0L 2= M(L) and 1L 2= J (L), 0 = + 1 if and only if either 0L 2 M(L) or 1L 2 J (L) (but not both), and 0 = + 2 if and only if 0L 2 M(L) and 1L 2 J (L). The following Proposition is known to hold for a wider class of lattices than distributive lattices (although not in general), see [3]. However, Lemma 3.7 allows us to give an intrinsically di erent proof in this special case which, since the proof is also succinct, we shall so do.
Proposition 5.1 Assume > 1. Then: (i) 0 + 2 . (ii) 0 = if and only if L is a chain.
Proof: (i) We have = jM (P )j = jA(P )j + jC (P )j + jB 0(P )j. Let fSi : 1 i ng be an enumeration of all nontrivial h -classes of P , and put i := jSij for i = 1; : : : ; n (hence i > 1). By Lemma 3.7(i) fp; qg \ SPi = ;whenever (p; q) 2 A, whence jAj Pn n i ? i i . Moreover, jC j = i 2 (if C = ;, then n = 0 and both sums vanish). Note that fp; qg 6 Si for i = 1; : : : ; n whenever (p; q) 2 B 0 . Writing 0 for jB 0j, we thus obtain =1
=1
?
n X i=1
2
i +
n X i=1
!
2 2i + 0
Note that 2 i 2i ? 2 for all i since i 2, hence 2 Combining, we obtain 2
?
n X i=1
i +
n X (2i i=1
17
i 2
? 1 ? 1=n) + 0
2i ? 1 ? 1=n.
or
?
n X i=1
i +
n X 2i i=1
? n ? 1 + 0
Now equals the number of antichains in P = J (L). We enumerate some of these antichains: there areS antichains with one element; further 0 antichains of 2 elements outside ni Si, namely the B 0 -pairs of P ; nally the number of antichains of at least 2 elements within each Si is 2i ? 1 ? i . Counting in the empty antichain, this gives the following estimate for : =1
n X 1 + + (2i i=1
or
+
n X 2i i=1
? 1 ? i ) + 0
?n?
n X
i=1 thus 0
i + 1 + 0
Comparing, we obtain ? 2 and + 2 , as claimed. (ii) If L is a chain, then obviously 0 = . Conversely, assume 0L = L. We argue inductively on L. By part (i), then, 0L = L +2 and so 1L 2 J (L) as well as 0L 2 M(L). Let u be the unique lower cover of 1L. If u 2 J (L), put L0 := L n f1Lg; then 0L = L since 0L = L, and, by induction, L is a chain. Alternatively u 2= J (L). We will show that this case cannot occur. Again put L0 := L n f1Lg, we see that now 0L = 0L ? 1 and L = L ? 1. Since 0L = 0L 2 M(L0) and 1L 2= J (L0), we conclude 0L = L + 1. Using the original assumption 0L = L, we obtain L = 0L ? 1 = 0L ? 2 = L ? 2 = L ? 1. But this contradicts L L ? 2 as proved in (i). 0
0
0
0
0
0
0
0
0
0
0
0
0
2
By Prop. 5.1, the number 0 of general sublattices of L is bounded from above by the number of elements of L. The next proposition shows that 0 can take almost all values below this bound:
Proposition 5.2 ? 0 takes all values except 1 as L varies over D01. Proof: By Proposition 5.1(ii), we have L ? 0L = 0 if and only if L is a chain.
18
Suppose L ? 0L = 1. Were 0L = L, it would follow that L ? L1, contradicting Proposition 5.1(i). Thus, 0L equals L + 1 or L + 2. In either case, 1L 2 J (L) or 0L 2 M(L), say 1L 2 J (L). Put L0 := L n f1Lg. Then L = L ? 1 and 0L = 0L ? 1. Thus L ? 0L = 1 which an inductive argument reveals is impossible. It remains to exhibit L with L ? 0L = for any any value 2. Suppose rst that is even, = 2 with 1. Put L = |2 2 {z : : : 2} copies We obtain 0L = 2 , L = 4 , thus L = 2 = as desired. Let now = 2 ? 1 with 2. Let L be 3 with the two doubly irreducible elements removed. Put L = L 2| 2 {z : : : 2} -2 copies Then 0L = 2( ? 2) + 4 = 2 and L = 4( ? 2) + 7 = 4 ? 1, thus L = 2 ? 1 = as desired. 0
0
0
2
2
2
2
1
1
0
2
2
2
2
De ne L := 0L=L. By 5.1(i) we have L 1 and 5.1(ii) says that L = 1 if and only if L is a chain. It is easy to see that, on the other hand, L may be made arbitrarily small: Indeed, consider the nite Boolean lattice Bn with n atoms whose dual poset An is an antichain with n elements. Since every pair of elements from An is a critical pair, we have 0Bn = n(n ? 1). On the other hand, since every subset of An is an order ideal, Bn = 2n. So Bn = n(n ? 1)=2n which can be made arbitrarily small by choosing n large enough. We can say somewhat more about the lower bound if we restrict the class of lattices considered. Let Dn be the class of all n?generated members of D01.
Corollary 5.3 For n 2, let L 2 Dn n Dn? . Then Fn L Cn (= 1), where Fn is the free distributive lattice on n generators and Cn is the n?element chain, the bounds being attained if and only if L = Fn or Cn, respectively. 1
19
Proof: Let L 2 Dn n Dn? . Choose a generating set G = fg ; : : : ; gng in L. Since L 62 Dn? , the sets Gi = G n fgig generate distinct proper sublattices Li L with gi 62 Li for 1 i n. For each i, choose a maximal sublattice Li Mi L; it is still the case that gi 62 Mi , so the Mi are pairwise distinct. It follows that 0L n. Since Fn is generated by its doubly irreducible elements, 0Fn = n. On the other hand, if L 6 = Fn, then jFnj > jLj. Hence, 0 for L 6= Fn, Fn = n=jFnj < L=jLj = L. 1
1
1
2
Let L 2 D01 with dual P = J (L). Consider an arbitrary critical pair (p; q) 2 M (P ). We write Lpq for the corresponding maximal sublattice O(Ppq ) of O(P ) as well as for its canonical copy within L. Our concern here is jL n Lpq j. By the de nition of Ppq , the down-sets of Ppq correspond to the down-sets of P which contain q whenever they contain p; accordingly, we will identify Lpq with fS 2 O(P ) : p 2 S ) q 2 S g. Consequently, L n Lpq = fS 2 O(P ) : p 2 S but q 2= S g. A general reference for the material in the remainder of this section is Rival [13, Section 4]. For q 2 P , de ne Sq 2 O(P ) by Sq := fx 2 P : x 6 qg. Sq is the largest down-set in P not containing q, and Sq is meet-irreducible in O(P ). The least down-set containing p is (p]; consequently, L n Lpq = fS 2 O(P ) : (p] S Sq g. Assume now that jL n Lpq j = 1. It is clear from the outset that in this case Lpq arises from L by the removal of a doubly irreducible element d 2 J (L) \ M(L); we want to relate this element to its associated critical pair (p; q) 2 M (P ).
Fact 5.4 The following are equivalent: (i) jL n Lpq j = 1.
(ii) p is doubly irreducible in L, and (p] [ [q) = P , (p] \ [q) = ;. (iii) L n Lpq = fpg.
Proof: (i)!(ii): By our description of L n Lpq we conclude that (p] = Sq , in other words, x p if and only if x 6 q. Since p 6 q as (p; q) is a critical pair, this establishes the asserted partition of P . Further, p is meet-irreducible in L with unique upper cover p _ q. 20
(ii)!(iii): For p and q as given, (p; q) 2 M with L n Lpq = (p]. (iii)!(i) is trivial.
2
Note that for p 2 L doubly irreducible there exists always a unique q 2 J (L) as required in 5.4(ii): Indeed, for each m 2 M(L) there exists a unique j 2 J (L) minimal with m 6 j since L is nite and distributive.
Proposition 5.5 The following are equivalent: (i) jL n Lpq j = 2. (ii) There exists a unique a 2 P which akp and akq. (iii) L n Lpq = fp; p _ ag. Proof: (i)!(ii): We have L n Lpq = f(p]; Sq g with Sq (p]. So there exists a 2 Sq n (p]. Then a > p is not possible, for otherwise a q, (p; q) being critical. Thus akp. The same argument excludes a < q, thus akq. If a0 2 P , a0kp and a0 kq, then a0 2 Sq n (p], thus (p] (p] [ (a0] Sq . We conclude a = a0 (and Sq = (p] [ fag, which we will need below). (ii)!(iii): The preceding arguments actually show that S ? q n (p] consists of elements which are not comparable to either p or q whenever S 2 O(P ), (p] S Sq . So (p][fag is the unique set of this type under the assumptions of (ii), and it corresponds to p _ a under the canonical isomorphism L = O(J (L)). (iii)!(i) is trivial. 2
Note that for a; p; q satisfying say 5.5(ii) we have p 2= M(L), p _ a 2= J (L) and p _ a 2 M(L), so the interval [p; p _ a] satis es the requirements of [13, Theorem 3]. Indeed, working in O(P ), we see that (p] = ((p] [ (q]) \ ((p] [ (a]) (see (i)!(ii) above) which proves the rst assertion; the second is trivial while the third is justi ed by the fact that any down-set properly containing fp; ag must also contain q. 21
References [1] Abad, M., and Adams, M. E., The Frattini sublattice of a nite distributive lattice, Algebra Universalis 32 (1994), 314 { 329 [2] Adams, M. E., The Frattini sublattice of a distributive lattice, Algebra Universalis 3 (1973), 216 { 228 [3] Adams, M. E., Freese, R., Nation, J. B., and Schmid, J., Maximal sublattices and Frattini sublattices of bounded lattices, to appear [4] Balbes, R., and Dwinger, Ph., Distributive Lattices, University of Missouri Press, 1974, Columbia MO 65211 [5] Chen, C. C., Koh, K. M., and Tan, S. K., Frattini sublattices of distributive lattices, Algebra Universalis 3 (1973), 294 { 303 [6] Chen, C. C., Koh, K. M., and Tan, S. K., On the Frattini sublattice of a nite distributive lattice, Algebra Universalis 5 (1975), 88 { 97 [7] Faigle, U., and Turan, Gy., On the complexity of interval orders and semiorders, Discrete Math. 63 (1987), 131 { 141 [8] Fishburn, P. C., and Trotter, W. T., Posets with large dimension and relatively few critical pairs, Order 10 (1993), 317 { 328 [9] Hashimoto, J., Ideal theory for lattices, Math. Japon. 2 (1952), 149 { 186 [10] Kelly, D., and Trotter, W. T., Dimension theory for ordered sets, in: Ordered sets, I. Rival (ed.), Reidel Publ. Co., Dordrecht 1982, Holland, 171 { 211 [11] Koh, K. M., On the Frattini sublattice of a lattice, Algebra Universalis 1 (1971), 104 { 116 [12] Rabinovitch, I., and Rival, I., The rank of a distributive lattice, Discrete Math. 25 (1979), 275 { 279 [13] Rival, I., Maximal sublattices of nite distributive lattices, Proc. Amer. Math. Soc. 37 (1973), 417 { 420 22
[14] Ryter, Ch., and Schmid, J., Deciding Frattini is NP-complete, Order 11 (1994), 257 { 279
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