Join-Semidistributive Lattices and Convex Geometries

Advances in Mathematics 173 (2003) 1–49 http://www.elsevier.com/locate/aim

Join-semidistributive lattices and convex geometries

K.V. Adaricheva,*,1 V.A. Gorbunov,2 and V.I. Tumanov Institute of Mathematics of the Siberian Branch of RAS, Acad. Koptyug Prosp. 4, Novosibirsk, 630090 Russia

Received 3 June 2001; accepted 16 April 2002

Abstract

We introduce the notion of a convex geometry extending the notion of a ﬁnite closure system with the anti-exchange property known in combinatorics. This notion becomes essential for the different embeddingresults in the class of join-semidistributive lattices. In particular, we prove that every ﬁnite join-semidistributive lattice can be embedded into a lattice SPðAÞ of algebraic subsets of a suitable algebraic lattice A: This latter construction, SPðAÞ; is a key example of a convex geometry that plays an analogous role in hierarchy of join- semidistributive lattices as a lattice of equivalence relations does in the class of modular lattices. We give numerous examples of convex geometries that emerge in different branches of mathematics from geometry to graph theory. We also discuss the introduced notion of a strong convex geometry that might promise the development of rich structural theory of convex geometries. r 2002 Elsevier Science (USA). All rights reserved.

MSC: 06B05; 06B15; 08C15; 06B23

Keywords: Lattice; Join-semidistributive; Anti-exchange property; Convex geometry; Atomistic; Biatomic; Quasivariety; Antimatroid

A lattice ðL; 3; 4Þ is called join-semidistributive,if

x3y ¼ x3z implies that x3y ¼ x3ðy4zÞ for all x; y; zAL:

*Correspondingauthor. E-mail address: [email protected] (K.V. Adaricheva). 1 Current address: 134 South Lombard Ave., Oak Park, IL 60302, USA. 2 Deceased.

Join-semidistributive lattices were introduced in 1961 by Jo´ nsson [46] in his studies of free lattices. Nowadays, the class SD3 of join-semidistributive lattices, together with the class M of modular lattices, are amongthe most important extensions of D; the class of distributive lattices. (Observe that SD3-M ¼ D:) The investigation of join-semidistributive lattices originates in the study of objects such as free and lower bounded lattices as well as their sublattices [27] or lattices of quasivarieties [31,33]. Join-semidistributive lattices also have many applications in various branches of lattice theory and universal algebra such as the theory of varieties of lattices [44,48,57], tame congruence theory [28,42], and the theory of lattices with irredundant decompositions [17,32]. It is known [37] that a key to the study of modular lattices is given by combinatorial geometries, that is, modular geometric lattices and projective spaces. One of our main goals is to show that the latter geometrical objects have a counterpart in the theory of join-semidistributive lattices. This counterpart has already been christened convex geometries in combinatorics a longtime ago,and the antithesis with matroids clearly recognized. Namely, the role played by the exchange property for matroids is played by the anti-exchange property for convex geometries. More precisely, while matroids are lattices of closed sets of closure spaces with the exchange property, antimatroids are lattices of open sets of closure spaces with the anti-exchange property, see [15]. In the present paper, a convex geometry is a closure space with the anti-exchange property, see Definition 1.6. To our knowledge, the lattice-theoretical facet of convex geometries appears in full only in the present paper, especially as far as infinite convex geometries are concerned. This will justify further the often encountered juxtaposition of join-semidistributivity and modularity. The first source of finite convex geometries was found in lattice theory. In 1940, Dilworth [17] discovered these objects as a special class of those finite (lower) semimodular lattices that have unique irredundant decompositions. Later on, finite convex geometries were rediscovered many times as the abstract framework underlyingthe principal features of some combinatorial constructions (see a review of this topic in [56]). The systematical study of finite convex geometry as an abstraction of the notion of convexity in Euclidean space was launched by Jamison [43] and Edelman [20], and as the special greedoids by Korte and Lova´ sz [49]. For an update state of this theory, the reader should consult [21,50]. Accordingto one of the characterizations of finite convex geometries(see Theorem 1.9), a finite lattice is the closure lattice of some finite convex geometry iff it is join-semidistributive and lower semimodular. In Section 1, we shall prove (Theorem 1.11) that any finite join-semidistributive lattice can be embedded into some finite atomistic join-semidistributive lattice, or, equivalently, into the closure lattice of some finite atomistic convex geometry. Together with Theorem 1.4, this proves that the quasivariety SD3 of join-semidistributive lattices is generated by the closure lattices of finite convex geometries. Another source of ‘‘universal’’ join-semidistributive lattices is the class of lattices of quasivarieties of general algebraic systems, the latter as considered in [52] or [34]. Unlike algebras, these algebraic systems may contain relation symbols as well. K.V. Adaricheva et al. / Advances in Mathematics 173 (2003) 1–49 3

Due to a theorem by Gorbunov and Tumanov [36], any quasivariety lattice is embedded into the lattice SpðAÞ of algebraic subsets of some algebraic lattice A: The lattice SpðAÞ is, in turn, isomorphic to some quasivariety lattice. It is worth noting that SpðAÞ is represented as a subquasivariety lattice of the class of pure relational systems, see [35]. In Section 2, we prove (Theorem 2.4) that an analogous embedding result holds for arbitrary finitely presented join-semidistributive lattices. This, together with results proved in Section 1, implies that the quasivariety SD3 is generated by all quasivariety lattices. Observe that the statement obtained by replacinggeneral quasivariety lattices by lattices of quasivarieties of algebras does not hold, see [28,33]. In Section 3, we discuss the various possibilities of defininga strong convex geometry in the infinite case. We first consider various often encountered classes of convex geometries. It turns out that all these examples are atomistic, and that they have either an algebraic or a dually algebraic closure lattice. Further, we compare the previously proved embeddingtheorems with the analogousresults for combinatorial geometries. Among other results, we prove (Theorem 3.26) that any join- semidistributive lattice can be embedded into an atomistic, algebraic, biatomic convex geometry. Despite our initial intention to complete the current paper before the second author’s monograph [34] went into press, the order of the events has been reversed. As a consequence, the proofs of core results in Sections 1 and 2 in the present paper transcend their presentation given in the book. Theorems 1.11 and 2.4 were also announced without proofs in [65]. We use in this paper the same notation and terminology as in [37].

1. Join-semidistributive lattices and ﬁnite convex geometries

1.1. Local theorem for the class of join-semidistributive lattices

We will several times require the followingrelativized version of join- semidistributivity.

Deﬁnition 1.1. Let L be a lattice, let A be a subset of L: We say that L satisﬁes SD3ðAÞ; if x3y ¼ x3z implies that x3y ¼ x3ðy4zÞ for all xAL and all y; zAA:

3 In particular, L is join-semidistributive iff L satisﬁes SD3ðLÞ: We denote by A the set of ﬁnite (nonempty) joins of elements of A: The followingLemma was suggested by F. Wehrung as a generalization of our initial version of this statement.

Lemma 1.2. Let L be a lattice, let A be a subset of L. Then the following are equivalent:

(i) L satisﬁes SD3ðAÞ; 4 K.V. Adaricheva et al. / Advances in Mathematics 173 (2003) 1–49

(ii) the following holds: _ _ _ x3 yi ¼ x3 zj implies that x3 yi iom jon_ iom ¼ x3 ðyi4zjÞ iom; jon

for all xAL; m; n40; and y0; y; ymÀ1; z0; y; znÀ1AA: 3 (iii) L satisﬁes SD3ðA Þ: Proof. Implication ðiÞ)ðiiÞ proceeds exactly the same way as the proof of [27, Theorem 1.21]. Implications ðiiÞ)ðiiiÞ and ðiiiÞ)ðiÞ are trivial. &

We say that a lattice L is additively generated by a subset A if any element of L is a ﬁnite join of elements from A; that is, A3 ¼ L:

Corollary1.3. Let L be a lattice additively generated by a subset A. Then L is join- semidistributive iff L satisﬁes SD3ðAÞ:

Now we are ready to prove the followingembeddingtheorem.

Theorem 1.4. Any join-semidistributive lattice can be embedded into an ultraproduct of ﬁnite join-semidistributive lattices. In particular, the quasivariety SD3 is generated by its ﬁnite members.

Proof. Let L be an arbitrary join-semidistributive lattice and let M be a finite partial sublattice of L: Consider the meet-subsemilattice M4 of L generated by M and the join-subsemilattice M43 of L generated by M4: We shall prove that M43 is a finite join-semidistributive lattice with respect to the induced order, and that it contains M jMj as a partial sublattice. Evidently, jM4jp2 ; and the meet 0M of all elements of M jMj jM4j 2 is the least element of M4: Analogously, jM43jp2 p2 : As 0M is the least element in M43; the latter is a lattice in which the joins coincide with the joins in L and the meets of elements from M4 are the same as in L: It follows that M is embedded into M43 and M43 is additively generated by its subset M4: To prove that M43ASD3; we apply Corollary 1.3. If a3x ¼ a3y for some aAM43 and x; yAM4; then, by join-semidistributivity, a3x ¼ a3ðx4yÞ: As x; yAM4; we have x4y ¼ x4M y: Therefore, a3x ¼ a3ðx4M yÞ: A f Thus, any lattice L SD3 is locally embedded in the class SD3 of finite join- semidistributive lattices. By the local theorem due to Mal’cev, see [52, p. 164], L is f & embedded into some ultraproduct of systems from SD3:

By a finitely presented join-semidistributive lattice, we always mean a lattice defined within the class SD3 by a finite set of generators X and a finite set of relations in the variables X: Similarly, residual finiteness will be defined within the class SD3: K.V. Adaricheva et al. / Advances in Mathematics 173 (2003) 1–49 5

Corollary1.5. Any ﬁnitely presented join-semidistributive lattice is residually ﬁnite. Hence, the universal theory of SD3 is decidable.

The latter statement means that there is a recursive procedure that decides whether a given universal formula of lattice theory is valid in every join-semidistributive lattice. Equivalently, there is a recursive procedure that decides, for a given presentation

P1ð~aÞ¼Q1ð~aÞ; y; Pnð~aÞ¼Qnð~aÞð1:1Þ of a ﬁnitely presented join-semidistributive lattice L (where the Pi’s and the Qi’s are lattice polynomials), whether an equation Pð~aÞ¼Qð~aÞ holds in L: Hence, the word problem for the join-semidistributive lattice with presentation (1.1) is solvable.

Proof. The ﬁrst assertion follows from Theorem 3 in [12]. The second one follows from a general argument that probably ﬁrst appears in [53], see also [52]. &

Observe that in contrast with the results proved above for SD3; the variety of modular lattices is not generated by its ﬁnite members [25]. Moreover, the word problem is undecidable in free modular lattices FMðnÞ with nX4; see [26,39].

Problem 1. For a ﬁnite partial lattice P; is the word problem in FSD3 ðPÞ; the free join-semidistributive lattice on P; decidable in polynomial time?

By Corollary 1.5, the word problem in FSD3 ðPÞ is decidable. Moreover, it is well known that the word problem in ﬁnitely presented lattices, that is, in FLðPÞ for a ﬁnite partial lattice P; is decidable in polynomial time, see, for example, [27].

1.2. Finite convex geometries

In this section, we shall deﬁne convex geometries in general and discuss their basic properties in the ﬁnite case. For any set X; we denote by BðXÞ the set of all subsets of X: We recall that a map F : BðXÞ-BðXÞ is a closure operator on X if for all A; BAX: (1) ADFðAÞ; (2) ADB implies that FðAÞDFðBÞ; (3) FðAÞ¼FðFðAÞÞ:

A pair ðX; FÞ; where F is a closure operator on X; is said to be a closure space.We say that a closure space ðX; FÞ is * a zero-closure space,ifFð|Þ¼|; * atomistic,ifFðfxgÞ ¼ fxg for all xAX and Fð|Þ¼| (the latter condition is redundant unless X is a singleton).W * algebraic, if the equality FðAÞ¼ fFðUÞ : UDA ﬁniteg holds for all ADX: 6 K.V. Adaricheva et al. / Advances in Mathematics 173 (2003) 1–49

It is obvious that the set ClðX; FÞ of all closed subsets of a closure space ðX; FÞ; endowed with containment, forms a complete meet-subsemilattice of BðXÞ; that will be called the closure lattice of ðX; FÞ: Conversely, given a complete meet- subsemilatticeT L of BðXÞ; we can deﬁne a closure operator F on X by the rule FðAÞ¼ fYAL : ADYg; for all AABðXÞ: We say then that the closure space ðX; FÞ on X is induced by the system L of its closed sets. Notice that we always can assume that F separates points in X, i.e., FðfxgÞ ¼ FðfygÞ implies x ¼ y: We observe that in the atomistic case, the closure space can be recovered from the lattice, as the points of X correspond to the atoms of ClðX; FÞ: Hence, we shall often identify an atomistic closure space with its closure lattice.

Deﬁnition 1.6. A zero-closure space ðX; FÞ satisﬁes the anti-exchange property if the followingstatement holds:

xAFðA,fygÞ and xeA imply that yeFðA,fxgÞ

for all xay in X and all closed ADX: ðAEPÞ

We then say that ðX; FÞ is a convex geometry.

A standard example of a finite convex geometry is the following. Let X be a finite set of points in the n-dimensional Euclidean space En: For an arbitrary subset ADX; we define FðAÞ to be the intersection of X with the convex hull of A in En: Then n ðX; FÞ is a finite convexW geometry and its closure lattice is denoted by CoðE ; XÞ: A decomposition y ¼ fyi : ipng of an elementW yAL into a join of join-irreducible elements yiAL is called irredundant if y4 fyi : ipn; iajg for all jpn: Finally, a finiteV lattice is said to be locally distributive if for any xAL the interval ½x0; x with x0 ¼ fy : y!xg is a distributive lattice, where y!x means that x covers y: We now recollect a few known characterizations of finite convex geometries. The equivalence of statements (2)–(4) in the theorem below belongs to Edelman and Jamison [21] and the equivalence of (1) and (2) was proved by Dilworth [17]. The reference to other descriptions can be found in Monjardet [55,56].

Theorem 1.7. Let L be a ﬁnite lattice. Then the following are equivalent: (1) any element of L has a unique irredundant decomposition; (2) L is a locally distributive lattice; (3) L is the closure lattice ClðX; FÞ of a closure space ðX; FÞ with the anti-exchange property; (4) L is the closure lattice of a closure space ðX; FÞ with the property that for any closed subset AaX of X there exists xAX\A such that A,fxg is closed.

Property of a closure space ðX; FÞ in (4) above is often accepted as the deﬁnition of a convex geometry in combinatorics. We shall use later, at several places, the followingwell-known lemma. K.V. Adaricheva et al. / Advances in Mathematics 173 (2003) 1–49 7

Lemma 1.8. Let L be a ﬁnite join-semidistributive lattice. For all a, bAL such that a!b; there exists a least element p of L such that ppb and p4/ a; and p is join-irreducible. V Proof. Put X ¼fxAL : xpb and x4/ ag; so X is nonempty. Put p ¼ X: Then a3x ¼ b for all xAX; whence, by join-semidistributivity, a3p ¼ b; so p is the least element of X: In particular, p40: If p ¼ x3y with x; yop; then x3a ¼ y3a ¼ a; thus ppa; a contradiction. Hence, p is join-irreducible. &

The followingdescription of ﬁnite convex geometries seems to have appeared ﬁrst in explicit form in [19]. At least a part of the statement can be deduced from Theorem 1.7, but for the sake of completeness we give here the self-contained proof of this result.

Theorem 1.9 (Folklore). A ﬁnite lattice L is isomorphic to the closure lattice of some ﬁnite convex geometry iff L is join-semidistributive and lower semimodular.

Proof. Let ðX; FÞ be a ﬁnite convex geometry. Suppose that for some closed subsets A; B; C of X;

A3B ¼ A3C; but A0 ¼ A3ðB-CÞaA3B:

0 One can enumerate the set X\A ¼fx1; y; xng in such a way that the set Ai ¼ 0 0 0 0 0 A ,fx1; y; xig is closed for all ipn: As B ¼ B\A and C ¼ C\A are nonempty and 0 0 0 0 B -C ¼ |; there exists ion such that, say, B DAi but C D/ Ai; a contradiction. Thus, ClðX; FÞ is join-semidistributive. To prove the lower semimodularity of ClðX; FÞ; observe first that if A!B in ClðX; FÞ; then B ¼ A,fxg for some xAB\A: Indeed, if there exists xay in B\A; then B ¼ ClðA,fxgÞ ¼ ClðA,fygÞ; which contradicts (AEP). Now if A!A3B in ClðX; FÞ; then A3B ¼ A,fxg for some xAB\A; hence A-B!B: To prove the converse statement, consider a join-semidistributive and lower semimodular lattice L: Denote by JðLÞ the set of (nonzero) join-irreducibleT elements and put JðaÞ¼fxAJðLÞ : xpag; for all aAL: Evidently, JðAÞ" fJðaÞ : ADJðaÞg defines a closure operator on JðLÞ and LDClðJðLÞ; JÞ: We prove that ðJðLÞ; JÞ is a finite convex geometry. Consider a nonunit element aAL; and let bga: If bAJðLÞ; then JðbÞ¼JðaÞ,fbg: Otherwise the set C ¼ fcAL : cob; c4/ ag is nonempty. By Lemma 1.8, C has a least element c0 that belongs to JðLÞ: Suppose that C contains another element dAJðLÞ: Then a!a3d; ! hence a4d d by lower semimodularity, and a4d ¼ d * is a lower cover of d: As c0od we have also c0pd * pa; a contradiction. Thus, c0 is the unique element of C; and it is join-irreducible. Hence, JðbÞ¼JðaÞ,fc0g always holds. &

A lattice L is called atomistic if any nonzero aAL is the join of atoms. As any atomistic join-semidistributive lattice is lower semimodular (see Proposition 3.1), we get from Theorem 1.9 the following: 8 K.V. Adaricheva et al. / Advances in Mathematics 173 (2003) 1–49

Corollary1.10. Any ﬁnite atomistic join-semidistributive lattice is the closure lattice of some ﬁnite convex geometry.

We shall discuss in more detail further properties of convex geometries in Section 3, includingthe connection with combinatorial geometries.

1.3. Universality of ﬁnite atomistic convex geometries

Not all finite convex geometries are atomistic (in contrast with combinatorial geometries), see Theorem 1.9. It turns out that any finite convex geometry can be extended, both join- and meet- preservingly, to some finite atomistic convex geometry defined on the same underlyingset.

Theorem 1.11. Any finite join-semidistributive lattice L can be embedded into the closure lattice of some finite atomistic convex geometry defined on the set JðLÞ of join-irreducible elements of L. Furthermore, this embedding preserves the bounds.

By ‘‘preserves the bounds’’ we mean, of course, that a zero and a unit of L is mapped by this embeddingto a zero and a unit, respectively, of a convex geometry.

Proof. For any aAL; we put JðaÞ¼fxAJðLÞ : xpag: For all a; bAL such that a!b; it follows from Lemma 1.8 that there exists a least element pAL such that ppb and p4/ a; we denote this element by mða; bÞ: Furthermore, we put

Bða; bÞ¼fXDJðbÞ\JðaÞ : mða; bÞAX ) X ¼ JðbÞ\JðaÞg: ð1:2Þ

Claim 1. Let a, bAL such that a!b; let CABða; bÞ such that CaJðbÞ\JðaÞ: Then there exists xAJðbÞ\ðJðaÞ,CÞ such that C,fxgABða; bÞ:

Proof. From the deﬁnition of Bða; bÞ follows that mða; bÞeC: We put x ¼ mða; bÞ if JðbÞ\ðJðaÞ,CÞ¼fmða; bÞg; and we pick any xamða; bÞ in JðbÞ\ðJðaÞ,CÞ otherwise. Then x satisﬁes the required conditions. &

For xAL; we shall construct, by induction on x; a subset Tx of BðJðLÞÞ: | (T0) We put T0L ¼f g: (T1) For xAJðLÞ; we put

Tx ¼ Tx ,fB,fxg : BATx g; * *

where x * denotes the unique lower cover of x: K.V. Adaricheva et al. / Advances in Mathematics 173 (2003) 1–49 9

(T2) For xAL\ðJðLÞ,f0LgÞ; we put

Tx ¼ Tox,fJðyÞ,C : y!x; CABðy; xÞg; S where we put Tox ¼ yox Ty and Bðy; xÞ is deﬁned as in (1.2).

A Hence, all Tx; for x L; are subsets of BðJðLÞÞ: We put T ¼ T1L : Further elementary properties of the sets Tx are summarized in the following claim:

Claim 2. The following assertions hold:

(i) xpy implies that TxDTy; for all x; yAL; (ii) fxgATx for all xAJðLÞ; (iii) JðxÞ is the largest element of Tx for all xAL:

Claim 3. Let x; yAL such that xoy: Then there exists zAJðyÞ\JðxÞ such that JðxÞ,fzgATy:

Proof. We ﬁrst replace y by any y0AL such that x!y0py; so we see that without ! A loss of generality, we may assume that x y: If y JðLÞ; then x ¼ y * ; thus, by Claim 2(iii), JðxÞ,fyg¼JðyÞATy; so we may take z ¼ y: Suppose now that yeJðLÞ: By Claim 1, there exists zAJðyÞ\JðxÞ such that fzgABðx; yÞ: Then JðxÞ,fzgATy: &

Claim 4. For all BAT with BaJðLÞ; there exists xeB such that B,fxgAT:

Proof. The case where B ¼ JðaÞ for some aAL is taken care of by Claim 3. Thus, to conclude the proof of Claim 4, it sufﬁces to prove that for any xAL and any \ a \ BATx Tox; if B JðxÞ; then there exists yAJðxÞ B such that B,fygATx: We argue by induction on x: The statement holds vacuously if x ¼ 0L: Now \ a suppose that x40L; and let BATx Tox such that B JðxÞ: We consider two cases. Case 1. xeJðLÞ: So B ¼ JðyÞ,C for some y!x and some CABðy; xÞ: From BaJðxÞ follows that CaJðxÞ\JðyÞ: By Claim 1, there exists zAJðxÞ\ðJðyÞ,CÞ such that C,fzgABðy; xÞ: Hence zeB and B,fzg¼JðyÞ,ðC,fzgÞATx: Case 2. xAJðLÞ: Then B ¼ C,fxg for some CATx : Let ypx be a minimal * * \ element of L such that CATy: Hence CATy Toy:

* If CaJðyÞ; then, by the induction hypothesis, there exists zAJðyÞ\C such that C,fzgATy: Therefore, zAJðxÞ\B and B,fzg¼ðC,fzgÞ,fxgATx: * Finally, suppose that C ¼ JðyÞ: If y ¼ x * ; then B ¼ Jðx * Þ,fxg¼JðxÞ; a A \ contradiction. Hence yox * : By Claim 3, there exists z Jðx * Þ JðyÞ such that C,fzgATx : So zeB; and B,fzg¼ðC,fzgÞ,fxg belongs to Tx: * This concludes the proof of Claim 4. & 10 K.V. Adaricheva et al. / Advances in Mathematics 173 (2003) 1–49

Claim 5. Let SðTÞ be the meet-subsemilattice of BðJðLÞÞ generated by T. Then the closure space on JðLÞ induced by SðTÞ is a ﬁnite atomistic convex geometry.

Proof. The ‘‘atomistic’’ part of the statement follows immediately from Claim 2(ii). Now we verify the ‘‘convex geometry’’ statement. For any XDBðJðLÞÞ; we put 1 Xð0Þ ¼ X; Xð1Þ ¼fB-C : B; CAXð0Þg; and Xðkþ1Þ ¼ðXðkÞÞð Þ for all kX0: As JðLÞ is ﬁnite, SðTÞ¼T ðnÞ for some n: Hence, in view of Claim 4, it sufﬁces to prove that the assertion ‘‘for any BAX; BaJðLÞ; there exists xeB such that B,fxgAX’’ can be lifted from X to Xð1Þ: Let X have this property, and let D ¼ B-C for some B; CAX: We may assume without loss of generality that B\Da|: By assumption, the set JðLÞ\C ¼fy1; y; ykg can be enumerated in such a way that C,fy1; y; yigAX for all iAf1; y; kg: Observe that B\D is a nonempty subset of fy1; y; ykg: Let ym be the element of B\D with the smallest index. Then y1; y; ymÀ1eB\D; hence B-ðC,fy1; y; ymgÞ ¼ ð1Þ D,fymgAX ; and we are done. &

It remains to prove that L is embedded in SðTÞ via the rule a/JðaÞ: Evidently, the map x/JðxÞ is a one-to-one meet-homomorphism from L into SðTÞ: Let us prove that Jðx3yÞ¼JðxÞþJðyÞ; where þ denotes the join in SðTÞ; for all x; yAL: As any element in SðTÞ is a meet of elements from T; it suffices to prove that, for any BAT; if JðxÞ,JðyÞDB; then Jðx3yÞDB: The result is trivial if x and y are comparable, so, suppose that x and y are incomparable, in notation, xjjy: Let zAL be minimal such that BATz: In particular, BDJðzÞ; so from xjjy follows that x; yoz: We now argue by cases. Case 1: zAJðLÞ: Then fzgDBDJðzÞ: We argue by induction on jBj: From x; yoz follows that x; ypz ; thus, by the definition of Tz; B ¼ C,fzg for some CATz : * * Therefore, JðxÞ,JðyÞDC; so, by the induction hypothesis, Jðx3yÞDCDB: Case 2: zeJðLÞ: Then JðtÞDBDJðzÞ for some t!z: If JðxÞ,JðyÞDJðtÞ; then x3ypt; whence Jðx3yÞDJðtÞDB: Thus, we may now suppose that JðyÞD/JðtÞ; that is, y4/ t: Observe that ypz: Hence, mðt; zÞpy; so mðt; zÞAJðyÞDB; from which it follows, by the definition of Tz; that B ¼ JðzÞ; so Jðx3yÞDB: &

The proof of Theorem 1.11 is completed. &

Remark 1.12. It is well known that the modular analogue of Theorem 1.11 fails. For example, let K be the subgroup lattice of ðZ=4ZÞ3; so K is modular. Moreover, K is subdirectly irreducible, hence it follows easily from Frink’s embeddingTheorem and the results of Herrmann and Huhn [40] that K cannot be embedded into any complemented modular lattice. Hence, a fortiori, it cannot be embedded into any ﬁnite atomistic modular lattice.

There is a unique ‘‘largest’’ finite convex geometry extending a given finite convex geometry ðX; jÞ: We shall call a finite convex geometry ðX; c1Þ an extension of a finite convex geometry ðX; c0Þ defined on the same underlyingset X; if ClðX; c0Þ is a K.V. Adaricheva et al. / Advances in Mathematics 173 (2003) 1–49 11 sublattice of ClðX; c1Þ: The extensions of a given finite convex geometry ðX; c0Þ can be ordered as follows: if ðX; c1Þ and ðX; c2Þ are two such extensions, we say that ðX; c1ÞpðX; c2Þ if c2ðYÞDc1ðYÞ for any YDX; equivalently, ClðX; c1ÞDClðX; c2Þ:

Corollary1.13. Any ﬁnite convex geometry ðX; jÞ has a largest extension. Moreover, this extension is atomistic.

We observe here that the construction of Theorem 1.11 does not give the largest ﬁnite convex geometry extending the given lattice L: See further details in Remark 1.15.

Proof. Put L ¼ ClðX; jÞ: Since jð|Þ¼|; j separates points and satisfies the anti- exchange property, the rule x/jðfxgÞ defines an isomorphism from ðX; jÞ onto ðJðLÞ; JÞ: Let ðX; c1Þ; y; ðX; cnÞ be all the different extensions of ðX; jÞ: It follows from the previous paragraph and from Theorem 1.11 that one of the extensions ðX; c1Þ; y; ðXT; cnÞ is atomistic. Now we define a closure operator c on X by the D rule cðYÞ¼ ipn ciðYÞ; for all Y X: It is proved in [20] that ðX; cÞ is a finite convex geometry. It is not hard to prove that ðX; cÞ is an extension of ðX; jÞ as well. Indeed, if a3b ¼ c in ClðX; jÞ then a3b ¼ c in ClðX; ciÞ for any ipn; hence a3b ¼ c in ClðX; cÞ: Thus, ClðX; cÞ coincides with one of ðX; c1Þ; y; ðX; cnÞ and ClðX; ciÞDClðX; cÞ; for all ipn: Since one of the closure spaces ðX; ciÞ is atomistic, ðX; cÞ is atomistic as well. &

From Theorems 1.4 and 1.11, we also get

Corollary1.14. The quasivariety SD3 is generated by the closure lattices of ﬁnite atomistic convex geometries.

Remark 1.15. Lower bounded lattices form a proper subclass of join-semidistributive lattices (see definition and further details in Section 3). There is a standard way to embed a finite lattice into a finite atomistic lattice, see [64], and this procedure preserves lower boundedness. We recall here the main features of this construction. Let L be a finite lattice. A subset UDJðLÞ is called a minimal cover of an element vAJðLÞ if _ _ \ A vp U and v4/ u * 3 ðU fugÞ for all u U:

Consider the lattice AðLÞ whose elements are the sets XDJðLÞ that contain vAJðLÞ whenever some minimal cover of v is a subset of X: Clearly, AðLÞ is an atomistic lattice with atoms fvg; vAJðLÞ: Moreover, the rule a/JðaÞ deﬁnes a lattice embeddingof L into AðLÞ; see [64]. Let us prove that this embeddingdoes not preserve the join-semidistributive law as a rule. Consider the lattice L depicted in Fig. 1. It is easy to verify that L is 12 K.V. Adaricheva et al. / Advances in Mathematics 173 (2003) 1–49

a d e b

0 Fig. 1. join-semidistributive, JðLÞ¼fa; b; c; d; eg; and AðLÞ is deﬁned by the following relations:

apb3e; bpa3d; a3bpc3d3e:

In particular, it follows that

a3ðd3eÞ¼b3ðd3eÞad3e; whence AðLÞ is not join-semidistributive.

Remark 1.16. In contrast with the construction presented above, the embedding of a finite join-semidistributive lattice into the atomistic join-semidistributive lattice constructed in Theorem 1.11 does not preserve lower boundedness as a rule. For a poset P; let CoðPÞ denote the lattice of convex subsets of P; as defined in Section 3.1. Let 4 denote the four-element chain. Then Co 4 is the lattice defined by % ð%Þ the generators (atoms) a; b; c; d and the relations bpa3c; cpb3d; and b; cpa3d: Let L Co 4 \ a; d : Then LD3 3; and it is easy to verify that the construction of ¼ ð%Þ f g % Â % Theorem 1.11 embeds L into Co 4 ; the latter known not to be lower bounded. Since ð%Þ the largest extension of L is the Boolean lattice Bð4Þ; we also conclude that the construction of Theorem 1.11 does not provide the largest extension for an arbitrary join-semidistributive lattice. Still, both the largest extension and the extension provided by the proof of Theorem 1.11 are intrinsic, for example, every automorphism of L extends to an automorphism of the extension. This property is a priori not satisfied by the construction given in the proof of Theorem 4.3.3 in [34], which depends of the initial choice of a linear extension of the partial orderingof L: K.V. Adaricheva et al. / Advances in Mathematics 173 (2003) 1–49 13

2. Lattices of algebraic subsets

Let L be a complete lattice. A subset XDL is called algebraic,ifX is closed under Varbitrary meets and under joins of nonempty up-directed subsets. In particular, | ¼ 1LAX: Note that if L is an algebraic lattice, then X; endowed with the induced ordering, is also an algebraic lattice. We shall also need the following classical lemma aboutW algebraicW lattices. We say that a complete lattice L is upper continuous,if A a4 iAI xi ¼ iAI ða4xiÞ for any a A; any up-directed poset I; and any increasing family ðxiÞiAI of elements of A: Let lower continuous lattices be deﬁned dually. We say that a lattice L is weakly atomic, if for all aob in A; there are x; yAA such that apx!ypb: We say that L is spatial, if every element of L is a join of completely join-irreducible elements of L:

Lemma 2.1. Every algebraic lattice A satisﬁes the following properties:

(i) A is upper continuous. (ii) A is weakly atomic. (iii) A is dually spatial.

Proof. (i) See [13, Lemma 2.3]. (ii) See [13, Lemma 2.2], or [34, Exercise 1.3.1]. (iii) See [13, Proposition 6.1], or [30, Theorem I.4.22]. Also [34, Lemma 1.3.2]. &

Let SpðLÞ denote the lattice of algebraic subsets of L ordered by containment. It is easy to prove that SpðLÞ is dually isomorphic to the lattice of continuous closure operators deﬁned on L (see [34, Proposition 1.3.10]). The lattices of algebraic subsets emerged in the paper by Gorbunov and Tumanov [35] where the authors prove that for any algebraic lattice A the lattice SpðAÞ is isomorphic to LqðKÞ for some quasivariety K of relational structures. In particular, SpðAÞ is join-semidistributive and dually algebraic. It turns out that the conclusion of join-semidistributivity can be reached under weaker assumptions on the lattice A; see also [33].

Theorem 2.2. Let A be a complete lattice. Then the following assertions hold:

(i) If A is upper continuous, then SpðAÞ is join-semidistributive and lower continuous. (ii) If A is algebraic, then SpðAÞ is dually algebraic.

Proof. (i) See [34, Theorem 4.1.1]. (ii) See [34, Theorem 5.6.7]. &

For arbitrary complete A; some additional information can be found in Proposition 3.15.

Example 2.3. Let A ¼½0; 1 be the real unit interval. Then SpðAÞ consists of all topologically closed subsets of ½0; 1 that contain 1 as an element. Hence, f1g is the 14 K.V. Adaricheva et al. / Advances in Mathematics 173 (2003) 1–49 only dually compact element of SpðAÞ; so SpðAÞ is not dually algebraic, although A is upper continuous.

The principal aim of Section 2 is to prove the followingstatement.

Theorem 2.4. Let L be a join-semidistributive lattice deﬁned, within SD3; by ﬁnitely many relations. Then there exists an algebraic and dually algebraic lattice A such that L embeds into SpðAÞ; via an embedding that preserves the existing bounds.

Remark 2.5. Let LqðKÞ denote the class of all subquasivarieties of a given quasivariety K: It is proved in [33] that the ‘‘algebraic’’ part of conclusion of Theorem 2.4 also holds for all lattices of the form LqðKÞ; that is, there exists a 0; 1- embeddingof LqðKÞ into SpðAÞ for some algebraic lattice A:

Remark 2.6. As any free lattice is join-semidistributive (and deﬁned by the empty set of relations!), Theorem 2.4 yields that any free lattice embeds into SpðAÞ for some algebraic and dually algebraic lattice A.

Another class of lattices to which Theorem 2.4 applies is the class of all ﬁnitely presented (within SD3) join-semidistributive lattices, hence any ﬁnitely presented join-semidistributive lattice has a 0; 1-embedding into SpðAÞ for some algebraic and dually algebraic lattice A. Together with the result of [35] mentioned above, this gives the following

Corollary2.7. The quasivariety SD3 of join-semidistributive lattices is generated by the class of lattices of quasivarieties.

Observe that the class of lattices of quasivarieties of algebras generates a proper subquasivariety of SD3; see [28,31]. The proof of Theorem 2.4 is based on the next two statements havingan interest of their own.

Theorem 2.8. For any lattice L additively generated by a subset A such that 0eA; there exists a dually algebraic lattice TðL; AÞ such that L is embedded into SpðTðL; AÞÞ: Furthermore, this embedding preserves the existing bounds.

Theorem 2.9. If L is a ﬁnite join-semidistributive lattice and A is the set of atoms of L, then TðL; AÞ is both algebraic and dually algebraic.

Remark 2.10. It follows from Theorems 1.11, 2.8 and 2.9 that every finite join-semidistributive lattice L embeds into SpðAÞ for some algebraic lattice A. This lattice A is, in general, infinite, although L is finite. The fact that A has to be infinite is fully justified by Theorem 3.27, that states that A can be taken finite iff L is lower bounded. K.V. Adaricheva et al. / Advances in Mathematics 173 (2003) 1–49 15

This strange phenomenon contrasts with other representation results such as the one of Pudla´ kandTuma,’ see [60], that states that every ﬁnite lattice embeds into some ﬁnite partition lattice.

The construction of the lattice TðL; AÞ is given in Section 2.1 and the proofs of Theorems 2.8 and 2.9 are given in Sections 2.2 and 2.3, respectively. Before provingTheorem 2.4, we state a lemma:

Lemma 2.11. Let ðAiÞiAI be a family of algebraic (and dually algebraicQ ) lattices. Then there exists an algebraic (and dually algebraic) lattice A such that iAI SpðAiÞ has a 0; 1-embedding into SpðAÞ: Q Proof (see also Gorbunov [35]). We ﬁrst observe that the product A ¼ iAI Ai is an algebraic (and dually algebraic) lattice. We can deﬁne a map Y f : SpðAiÞ-SpðAÞ iAI by the rule Y Y A f ððXiÞiAI Þ¼ Xi for all ðXiÞiAI SpðAiÞ: iAI iAI

It is not hard to verify that f is a 0; 1-embeddingof lattices. &

Proof of Theorem 2.4. Let L be a lattice defined within SD3 by finitely many relations. Then, by Theorem 2.4.1 in [34], this lattice is limit-projective in SD3; which, in particular, implies that if L is embedded into a superdirect limit of a direct spectrum of lattices from SD3; then it is embedded into some lattice from that direct spectrum. In view of Theorem 2.3.6 in [34], the quasivariety QðKÞ generated by a class K coincides with LsPsðKÞ; where Ls and Ps are the operators of taking superdirect limits and subdirect products, respectively. Since the quasivariety SD3 is generated by its finite members (Theorem 1.4), it follows that there exists a subdirect decomposition of L of the form Y L+ Li; for finite join-semidistributive lattices Li; iAI: ð2:1Þ iAI

Observe, in particular, that the embeddingof (2.1) preserves the existingbounds. In view of Theorem 1.11, Theorems 2.8 and 2.9 imply that any ﬁnite join- semidistributive lattice can be 0; 1-embedded into SpðAÞ for some algebraic (and dually algebraic) lattice A: Applyingthis to Li; for iAI; yields an algebraic (and duallyQ algebraic) lattice Ai such that Li has a 0; 1-embeddinginto SpðAiÞ: By Lemma 2.11, iAI SpðAiÞ can be 0; 1-embedded into SpðAÞ for a suitable algebraic (and dually algebraic) lattice A: & 16 K.V. Adaricheva et al. / Advances in Mathematics 173 (2003) 1–49

More information about the lattices SpðAÞ can be found in Section 3.6. We now turn to the proofs of Theorems 2.8 and 2.9.

2.1. Principal construction

Let ðL; þ; ÁÞ be an arbitrary lattice and let A be a ﬁxed subset in L that additively generates L: Note that until the end of the proof we will be using þ; Á to denote the operations of join and meet on the lattice L in order to distinguish them from those on the lattice TðL; AÞ that we shall construct. We shall consider the set RðAÞ whose elements are the nonempty words a ¼ /a1; a2; y; anS on the alphabet A that satisfy the condition

ai4/ aiþ1; for all ipn À 1: ð2:2Þ

We shall put jaj¼n; the length of a: Greek letters will always denote words or segments of words, that is, one or several consecutive letters of a word, while roman font will be kept for letters from A: By eðaÞ we shall denote the last letter an in a and the remainder of the word will be denoted by a%: Thus a ¼ a%_/eðaÞS; where _ stands for concatenation of sequences. We define a partial order L on RðAÞ setting a ¼ /a1; a2; y; amSL / y S ? b1; b2; ; bn ¼ b iff mpn and there exist 1pj1o ojmpn such that ai ¼ bji for all iAf1; y; mg: If jm ¼ n then we say that b dilutes a; in notation aLdilb; and if _ b ¼ a g for some g then we say that b extends a; in notation aLextb: If aLextb and aab; we write aKextb: If aLb and aab; we write aKb: We extend the symbol K to RðAÞ,f|g by statingthat |Ka; for any aARðAÞ: It is clear that fbARðAÞ : bLag is finite, for each aARðAÞ: For a subset X of RðAÞ; we denote by Min X the set of minimal elements of X with respect to L: For any kX1 we define a partial ðk þ 1Þ-ary operation fk on RðAÞ as follows:

a ¼ fkða0; y; akÞ; if a ¼ a0 ¼ ? ¼ ak and eðaÞpeða0Þþ? þ eðakÞ:

A subset FDRðAÞ will be called an L-filter in RðAÞ if it is an order-filter in ðRðAÞ; LÞ which is closed with respect to the fk’s, that is, aAF whenever kX1; a0; y; akAF; and a ¼ fkða0; y; akÞ: In particular, both | and RðAÞ are L-filters in RðAÞ: Evidently, the set of all L-filters in RðAÞ forms a complete lattice TðL; AÞ¼ ðT; 3; 4Þ with respect to the reverse inclusion.

Lemma 2.12. For any aARðAÞ; the set RðaÞ¼fbARðAÞ : aLbg is an L-ﬁlter.

Proof. Let aLa0; y; ak: We have to prove that aLfkða0; y; akÞ: So a0 ¼ ? ¼ ak and eða0Þpeða0Þþ? þ eðakÞ: If eðaÞ¼eðajÞ for all jpk then eða0Þpeða0Þ; which contradicts (2.2). Hence aLa0: & K.V. Adaricheva et al. / Advances in Mathematics 173 (2003) 1–49 17

For XDRðAÞ and nAo; we deﬁne a subset X ðnÞ of RðAÞ by X ð0Þ ¼ X and X ðnþ1Þ ¼ ðX ðnÞÞð1Þ for any n; with

X ð1Þ ¼ X,faARðAÞ : (kX1;

(a0; y; akAX such that fkða0; y; akÞLag:

ðnÞ Evidently,S the sequence ðX ÞnAo is increasingwith respect to inclusion. We put % ðnÞ X ¼ nAo X : Observe the followingobvious fact.

D % Lemma 2.13. For any X RðAÞ; the set X is the smallestV L-ﬁlter in RSðAÞ containing X. In particular, for any family ðFiÞiAI in TðL; AÞ; we get iAI Fi ¼ iAI Fi: S It is obvious that X ð1Þ ¼ fY ð1Þ : YDX; jYjoog for any XDRðAÞ: We get by induction that the analogous equality holds for X ðnÞ; for any n: Hence, the closure operator X/X% on RðAÞ is algebraic, which implies the following observation:

Lemma 2.14. The lattice TðL; AÞ is dually algebraic.

2.2. Proof of Theorem 2.8

In this subsection, we assume that L is a lattice, additively generated by a subset A of L; and that 0eA: Recall that by Min F we denote the set of minimal elements of a given subset FDRðAÞ: For any aAL; we set

cðaÞ¼fFATðL; AÞ : eðaÞpa for all aAMin Fg:

It is obvious that if L has a zero, then cð0LÞ¼f|g is the zero of SpðTðL; AÞÞ; while if L has a unit, then cð1LÞ¼TðL; AÞ is the unit of SpðTðL; AÞÞ: Hence, c preserves the existingbounds.

Lemma 2.15. The set cðaÞ is an algebraic subset of TðL; AÞ:

Proof.W Let fFi T: iAIg be an arbitrary up-directed subset in cðaÞ: We shall prove that F ¼ iAI Fi ¼ iAI Fi belongs to cðaÞ: Let a in Min F: There are only ﬁnitely many elements less than a in the poset ðRðAÞ; LÞ; hence a is minimal in Fj for some jAI: This yields that eðaÞpa: A Now considerV an arbitrary family fFi : Si Ig of ﬁlters in cðaÞ: We have to check that F F belongs to c a : As F F ; it is enough to prove that e a a ¼ iAI i ð Þ ¼S iAI i ð Þp ðnÞ A holds for any minimal element a of ð iA FiÞ ; for any n o: We proceed by induction on n: 18 K.V. Adaricheva et al. / Advances in Mathematics 173 (2003) 1–49

n e a n For ¼ 0 we have ðaÞp by assumption.S Suppose that the statement holdsS for and let a be a minimal element of ð F Þðnþ1Þ: We may assume that aeð F ÞðnÞ: iAI i S iAI i 3 A ðnÞ By deﬁnition, a fkðb0; y; bkÞ for some kX1 and b0; y; bk ð iAI FiÞ : It follows a from the minimality of a that a ¼ fkðb0; y; bkÞ: By assumption, a b0: Hence, a ¼ b ¼ ? ¼ b and eðaÞpeðb Þþ? þ eðb Þ: It remains to prove that eðb Þpa for all 0 k 0 k S j A ðnÞ jpk: For all j f0; y; kg; there exists a minimal element gj of ð iAI FiÞ such that g Lb : If eðb Þaeðg Þ then g Lb ¼ a: This implies that g ¼ a as a is a minimal j j jS j j j j element of F ðnþ1Þ; and we get a contradiction with the assumption S ð iAI iÞ e ðnÞ a ð iAI FiÞ : Consequently, eðbjÞ¼eðgjÞ; so, by the induction hypothesis, eðgjÞpa: Hence eðaÞpa: &

Lemma 2.16. The map c is an isomorphic embedding of L into SpðTðL; AÞÞ:

Proof. Evidently, c preserves the meets. It follows that cðaÞ3cðbÞDcða þ bÞ: Now we prove the converse inclusion. Consider FAcða þ bÞ: By Lemma 2.12, [ ^ F ¼ fRðgÞ : gAMin Fg¼ fRðgÞ : gAMin Fg; where Min F is the set of minimal elements of F: Thus it is enough to prove RðgÞAcðaÞ3cðbÞ for any gARðAÞ with eðgÞpa þ b: If eðgÞpa or eðgÞpb; then RðgÞAcðaÞ or RðgÞAcðbÞ; respectively, and we are done. Now suppose that eðgÞ4/ a; b: Express a ¼ x0 þ ? þ xk and b ¼ y0 þ ? þ ys as joins of elements from A; so, in particular, eðgÞ4/ xi and eðgÞ4/ yj for all i; j: Then _ _ g /xiS and g /yjS belongto RðAÞ; and

_ _ _ _ g ¼ fkþsþ1ðg /x0S; y; g /xkS; g /y0S; y; g /ysSÞ since eðgÞpa þ b ¼ x0 þ ? þ xk þ y0 þ ? þ ys: It follows that ^ ^ _ _ RðgÞ¼ ðRðg /xiSÞ : i ¼ 1; y; kÞ4 ðRðg /yjSÞ : j ¼ 1; y; sÞ:

_ _ Clearly, Rðg /xiSÞAcðaÞDcðaÞ3cðbÞ and Rðg /yjSÞAcðbÞDcðaÞ3cðbÞ; whence RðgÞAcðaÞ3cðbÞ: Finally, c is one to one. Indeed, if a4/ b and a ¼ x0 þ ? þ xk with xiAA then xi4/ b for some ipk: Put a ¼ /xiS: Then RðaÞAcðaÞ and RðaÞecðbÞ: &

2.3. Proof of Theorem 2.9

We shall start with general considerations that involve neither ﬁniteness nor join- semidistributivity of L: K.V. Adaricheva et al. / Advances in Mathematics 173 (2003) 1–49 19

2.3.1. Minimal elements of X n and inferences In this subsection, let L be a lattice, let ADL: For XDRðAÞ and nAo; we deﬁne a subset X ½n of RðAÞ by X ½0 ¼ X and X ½nþ1 ¼ ðX ½nÞ½1 for any n; and the following

X ½1 ¼ X,faARðAÞ : (kX1;

(a0; y; akAX such that fkða0; y; akÞ¼ag:

½n Evidently,S the sequence ðX ÞnAo is increasingwith respect to inclusion. We put n ½n nD % X ¼ nAo X ; hence X X: The followingtwo lemmas will relate more precisely the subsets X n and X%; for any XDRðAÞ: It is trivial that X ðnÞ is an order-ﬁlter of ðRðAÞ; LÞ; for all XDRðAÞ and all nX1: The followinglemma is slightly less obvious:

½n Lemma 2.17. For any order-ﬁlter X of ðRðAÞ; LdilÞ and any nAo; the subset X is an order-ﬁlter of ðRðAÞ; LdilÞ:

Proof. It sufﬁces to prove the result for n ¼ 1: Let aAX ½1 and bARðAÞ such ½1 that aLdilb; we prove that bAX : If aAX; then this follows from the assumption on X: If aeX; then we can write a ¼ fkða0; y; akÞ for some kX1 and a0; y; akAX: _/ S Put bi ¼ b eðaiÞ ; for all iAf0; y; kg: From aLdilb follows that b ¼ fkðb0; y; bkÞ and that aiLdilbi for all i; so biAX by assumption on X: Therefore, bAX ½1: &

Lemma 2.18. Let X be an order-ﬁlter of ðRðAÞ; LÞ: Then Min X%DX n:

Proof. It is sufﬁcient to prove that Min X ðnÞDX ½n; for all nAo: We argue by induction on n: The result is trivial for n ¼ 0: Suppose that it holds for n; let aAMin X ðnþ1Þ; we prove that aAX ½nþ1: If aAX ðnÞ; then aAX ½n by the induction hypothesis, so aAX ½nþ1: Now suppose that aeX ðnÞ: Then, by the minimality of a; we ðnÞ can write a ¼ fkða0; y; akÞ; for some kX1anda0; y; akAX : ½n Hence, to conclude the proof, it sufﬁces to prove that aiAX for all ðnÞ iAf0; y; kg: By the induction hypothesis, this holds if aiAMin X : If ðnÞ ðnÞ aieMin X ; then there exists bAMin X such that bKai; so, by the induction ½n ðnÞ hypothesis, bAX : If bLa; then aAX ; a contradiction. Hence, bL/ a ¼ ai; so it ½n ½n follows from bLai that bLdilai; whence, since bAX and by Lemma 2.17, aiAX again. &

We shall now turn to the most important concept used in the proof n of Theorem 2.9. To prepare for it, we deﬁne a sub-function fk of fk; for all kX1: 20 K.V. Adaricheva et al. / Advances in Mathematics 173 (2003) 1–49

n Deﬁnition 2.19. For any kX1; we deﬁne a ðk þ 1Þ-ary partial operation fk on RðAÞkþ1 by the rule

n a ¼ fk ða0; y; akÞ; if a ¼ fkða0; y; akÞ and X eðaÞ4/ eðaiÞ; for all jAf0; y; kg: iaj

n A In particular, a ¼ fk ða0; y; akÞ implies that all the eðaiÞ; for i f0; y; kg; are distinct.

Definition 2.20. For a subset X of RðAÞ; let inferences from X be the nonempty, finite sequences of elements of RðAÞ built up accordingto the followingrules. (I0) /aS is an inference of a from X; for any aAX: (I1) If aARðAÞ\X; kX1; a0; y; akARðAÞ; I0; y; Ik are inferences of a0; y; ak; n respectively, from X; and a ¼ fk ða0; y; akÞ; then the finite sequence _ _?_ _/ S I ¼ I0 I1 Ik a ð2:3Þ is an inference of a from X:

We shall call expression (2.3) a canonical decomposition of I; and we shall call a the top of I: As we shall see later (see Corollary 2.25), such a decomposition is unique. If I ¼ /a0; y; anÀ1S; we shall denote by jIj¼n the length of I; and by rng I ¼ fa0; y; anÀ1g the range of I: The followingstraightforward lemma relates inferences and the correspondence X/X n:

Lemma 2.21. Let XDRðAÞ: Then the following assertions hold: (i) For any inference I from X, rng I is a subset of X n: (ii) Let aAX n: Then there exists an inference I of a from X.

We shall now establish various structural properties of inferences.

Lemma 2.22. Let XDRðAÞ; let aARðAÞ; let I be an inference of a from X. Then the following assertions hold: (i) All elements of rng I\fag are proper extensions of a: (ii) All entries of I are distinct (i.e., I is a one-to-one sequence).

Proof. Assertion (i) follows from an obvious induction argument on jIj: K.V. Adaricheva et al. / Advances in Mathematics 173 (2003) 1–49 21

(ii) We argue by induction on jIj: The result is trivial for jIj¼1: If jIjX2; we pick a canonical decomposition of I as in (2.3). Let b (resp., g) be the i-th entry (resp., the j-th entry) of I; with iaj: If b is the top of I; then, by (i), g is a proper extension of b: If b; g belongto the rangeof the same Is; with 0pspk; then bag by the induction hypothesis. Suppose that b (resp., g) belongs to the range of Ir (resp., Is) with ras: Then, by (i), b (resp., g) is an extension of ar (resp. as), however, araas by the n a deﬁnition of fk ; hence b g: &

It follows from Lemma 2.22(ii) that every word of RðAÞ occurs at most once in an inference I from X: If aArng I; then we say that a is a free word of I; if aAX; and an inner word, otherwise.

Definition 2.23. Let XDRðAÞ; let I and J be inferences from X: We say that I is a v _ _ subinference of J; in notation I J; if J ¼ I0 I I1 for some finite sequences I0 and I1: If I1 ¼ |; we say that I is a final segment of J: If rng I-rng J ¼ |; we say that I and J are disjoint, in notation I>J:

It follows from Lemma 2.22(ii) that if I v J; then the ﬁnite sequences I0 and I1 _ _ such that J ¼ I0 I I1 are unique.

Lemma 2.24. Let XDRðAÞ; let I and J be inferences from X. If I is a ﬁnal segment of J, then I ¼ J:

Proof. We say that I and J are comparable, if one is a final segment of the other. We argue by induction on jJj: The result is trivial for jJj¼1: Suppose now that jJjX2 _?_ _/ S and let J ¼ J0 Jn g be a canonical decomposition of J; with Js an inference of bs from X; for all sAf0; y; ng: From the definition of an inference follows that e a/ S _?_ _/ S g X; whence I g ; so I has a canonical decomposition I ¼ I0 Im g ; with Ir an inference of ar from X; for all rAf0; y; mg: Since I and J are comparable, Im and Jn are comparable, whence, by the induction hypothesis, Im ¼ Jn: An easy induction argument yields similarly that ImÀk ¼ JsÀk for all kAf0; y; minfm; ngg: Since I is a final segment of J; mpn and J can be decomposed as

_?_ _ _?_ _/ S J ¼ J0 JnÀmÀ1I0 Im g :

n n Hence, g ¼ fn ðb0; y; bnÀmÀ1; a0; y; amÞ¼fmða0; y; amÞ; whence (see Deﬁnition 2.19) m ¼ n: Therefore, I ¼ J: &

As easy consequences of Lemma 2.24, we record the two followingcorollaries:

Corollary2.25. Let XDRðAÞ; let I be an inference from X. Then the following assertions hold:

(i) For every aArng I; there exists a unique subinference J of I with top a: 22 K.V. Adaricheva et al. / Advances in Mathematics 173 (2003) 1–49

_?_ _/ S (ii) If jIjX2 and I ¼ I0 Ik g is a canonical decomposition of I, then, for any subinference J of I, either J ¼ I; or J v Is for some sAf0; y; kg: (iii) If jIjX2; then I has exactly one canonical decomposition.

If aArng I; we shall denote by IðaÞ the unique subinference of I with top a: Observe that a is a free word iff IðaÞ¼/aS: Observe also that the deﬁnition of IðaÞ is relative not only to I and a; but also to X: However, X will always be understood from the context.

Corollary2.26. Let XDRðAÞ; let I be an inference from X. Then the subinferences of I from X are nested, that is, for all subinferences J and K of I, either J v K; or K v J; or J>K:

Lemma 2.27. Let XDRðAÞ; let I be an inference from X. For any a; bArng I; the following are equivalent:

(i) IðbÞ v IðaÞ; (ii) rng IðbÞD rng IðaÞ; (iii) aLextb:

Proof. Implication ðiÞ)ðiiÞ: is trivial. ðiiÞ)ðiiiÞ: Assume that (ii) holds. Then bArng IðbÞDrng IðaÞ; whence, since IðaÞ is an inference of a from X and by Lemma 2.22(i), aLextb: ðiiiÞ)ðiÞ: Assume that aLextb; we prove that IðbÞ v IðaÞ; by induction on jIj: _ _ _/ S The conclusion is trivial if jIj¼1: If jIjX2; then let I ¼ Iðg0Þ ? IðgkÞ g be the canonical decomposition of I: If a ¼ g; then IðaÞ¼I and the conclusion is trivial. If b ¼ g; then, by Lemma 2.22(i), a ¼ b and the conclusion follows. Now suppose that both a; b are distinct from g: By Corollary 2.25(ii), there are r; v v v sAf0; y; kg such that IðaÞ IðgrÞ and IðbÞ IðgsÞ: If r ¼ s; then IðbÞ IðaÞ by the a induction hypothesis. Suppose that r s: By Lemma 2.22(i), gsLextb and grLextaLextb; thus one of gr and gs is an extension of the other, hence, since they have the same length, gr ¼ gs; a contradiction. &

Deﬁnition 2.28. Let XDRðAÞ; let I be an inference from X; let nAo; let a1; y; anArng I such that the subinferences Iða1Þ; y; IðanÞ are pairwise disjoint. We can decompose I as

_ _ _?_ _ _ I ¼ I0 Iðasð1ÞÞ I1 InÀ1IðasðnÞÞ In for finite sequences I0; y; In and a unique permutation s of f1; y; ng: We define a new finite sequence I½a1; y; an by

y _/ S_ _?_ _ / S_ I½a1; ; an¼I0 asð1Þ I1 InÀ1 asðnÞ In: K.V. Adaricheva et al. / Advances in Mathematics 173 (2003) 1–49 23

Observe that the deﬁnition of I½a1; y; an is relative not only to I and a1; y; an; but also to X: However, X will always be understood from the context.

Lemma 2.29. Let XDRðAÞ; let I be an inference from X, let nAo; and let a1; y; anArng I such that Iða1Þ; y; IðanÞ are pairwise disjoint. Then I½a1; y; an is an inference from X,fa1; y; ang:

_ _ Proof. It is sufﬁcient to prove the result for n ¼ 1: Hence, let I ¼ J0 IðaÞ J1; we _/ S_ prove that I½a¼J0 a J1 is an inference from X,fag: We may assume that a is an inner word of I: If J1 ¼ | then I½a¼/aS and we are done, so suppose that a| _ _ _/ S J1 : Let I ¼ Iðb0Þ ? IðbkÞ b be the canonical decomposition of I: By v Corollary 2.25(ii), there exists sAf0; y; kg such that IðaÞ IðbsÞ: By the induction hypothesis, IðbsÞ½a is an inference from X,fag: By Lemma 2.22(ii), all IðblÞ; for las in f0; y; kg; are inferences from X,fag: Therefore,

______/ S I½a¼Iðb0Þ ? IðbsÀ1Þ IðbsÞ½a Iðbsþ1Þ ? IðbkÞ b is an inference from X,fag: &

We now introduce a useful notation, inspired from the intuitive meaningof an inference. For XDRðAÞ; an inference I from X; kX1; a; a0; y; akArng I; let

A n I a ¼ fk ða0; y; akÞð2:4Þ be the statement that there exists a permutation s of f0; y; kg such that

_ _ _/ S IðaÞ¼Iðasð0ÞÞ ? IðasðkÞÞ a : ð2:5Þ

Deﬁnition 2.30. Let XDRðAÞ; let I be an inference from X; let b; gArng I: An I- inference chain from b to g is a collection a of ﬁnite sequences of the form

// S S a ¼ gi;j :0pjpki :0pipn for some nAo; kn ¼ 0; and k0; y; knÀ1X2; such that the followingrelations hold:

(i) b ¼ g0;0 and g ¼ gn;0; (ii) IAg ¼ f nðg ; y; g Þ for all i n: iþ1;0 ki i;0 i;ki o

If g ¼ f nðg ; y; g Þ; then g ¼ g : Hence, the followingLemma 2.31 is a iþ1;0 ki i;0 i;ki iþ1;0 i;0 somewhat more precise version of Lemma 2.22(i).

Lemma 2.31. Let XDRðAÞ; let gARðAÞ; let I be an inference of g from X, let bArng I: Then there exists an I-inference chain from b to g: 24 K.V. Adaricheva et al. / Advances in Mathematics 173 (2003) 1–49

Proof. We argue by induction on jIj: The conclusion is trivial if jIj¼1orb ¼ g; so suppose that jIjX2 and bag: Let _ _ _/ S I ¼ Iðg0Þ ? IðgkÞ g ð2:6Þ be the canonical decomposition of I: There exists sAf0; y; kg such that bArng IðgsÞ: Let s be any permutation of f0; y; kg such that sð0Þ¼s; and put di ¼ gsðiÞ; for all A A n i f0; y; kg: Then I g ¼ fk ðd0; y; dkÞ with d0 ¼ gs; which gives an I-inference chain from gs to g: Furthermore, by the induction hypothesis (applied to IðgsÞ), there exists an IðgsÞ-inference chain from b to gs: The juxtaposition of these two chains gives an I-inference chain from b to g: &

Corollary2.32. Let XDRðAÞ; let I be an inference from X. Then the set A of inner words of I is convex for the relation Lext; that is, aLextgLextb and a; bAA implies that gAA:

Proof. It follows from Lemma 2.27 that b and g belongto rng IðaÞ: Hence, by workinginside IðaÞ; we may suppose that I is an inference of a: By Lemma 2.31, there exists an I-inference chain from b to a; say,

(i) b ¼ g0;0 and a ¼ gn;0; (ii) IAg ¼ f nðg ; y; g Þ for all i n: iþ1;0 ki i;0 i;ki o

In particular, giþ1;0 ¼ gi;0 for all ion; and all gi;0; for 0pipn; are inner words of I: From aLextgLextb follows that there exists iAf0; y; ng such that g ¼ gi;0: &

The followingfundamental lemma makes it possible to use X n instead of X% in the proofs of most results in Section 2.3.2.

Lemma 2.33. Let XDRðAÞ; let gARðAÞ\X; let bARðAÞ such that RðbÞDX; let I be an inference of g from X. Then bLd implies that bLdild; for all dArng I:

Proof. We argue by induction on jIj: Since geX; jIjX2; so let _ _ _/ S I ¼ Iðg0Þ ? IðgkÞ g be the canonical decomposition of I: Let dArng I such that bLd: It follows from the a assumption on X that d g; so dArng IðgsÞ for some sAf0; y; kg: If gsAX; then / S % IðgsÞ¼ d ; hence, since bLd and bL/ g ¼ gs ¼ d; bLdild: If gseX; then, by the induction hypothesis (applied to IðgsÞ), bLdild again. &

2.3.2. TðL; AÞ for L finite atomistic We have already proved in Theorem 2.8 that TðL; AÞ is dually algebraic, for any arbitrary lattice L additively generated by a subset A: To prove that TðL; AÞ is also K.V. Adaricheva et al. / Advances in Mathematics 173 (2003) 1–49 25 algebraic if L is a finite join-semidistributive atomistic lattice with the set of atoms A; it is enough to prove that TðL; AÞ is upper continuous, see Section 2. Indeed, as TðL; AÞ is dually algebraic, it follows from Lemma 2.1(iii) that any of its elements is a join of completely join-irreducible elements, and the latter are compact in the presence of upper continuity. From now on until the end of the proof of Theorem 2.9, we shall fix a finite, join-semidistributive, atomistic lattice L; with set of atoms A; and we shall put p ¼jAj: Towards a contradiction, we shall suppose that TðL; AÞ is not upper continuous.

Lemma 2.34. There exist a ﬁlter FATðL; AÞ and words b; gARðAÞ such that jbjXjgjþ p and gAðF4RðbÞÞ\F:

Proof. Let upperW continuity fail at some XATWðL; AÞ and chain fFi : iAIgDTðL; AÞ; that is, X4 iAI Fi is properly contained in iAI ðXW4FiÞ: Then there exists g that A lies in X4Fi for arbitrary i I but not in X0 ¼ X4 iAI Fi: For any iAI; put Gi ¼ Fi\X0: Then X4Fi ¼ X0,Gi; the least L-filter containing X0,Gi: Thus, gAX0,Gi; for any iAI; and geX0: Now we want to find Gi all whose words are longenough.More specifically, weT need i0 such that all words in Gi0 have length | at least jgjþp: Such an i0 exists as iAI Gi ¼ and there are only finitely many words whose length is at most jgjþp À 1: A \ X Observe that g X0,Gi0 X0: Choose the least integer k 1 for which there are y A a1; a2; ; ak Gi0 such that

gAX04Rða1Þ4?4RðakÞ:

Now if we denote F ¼ X04Rða1Þ4?4RðakÀ1Þ and b ¼ ak then

gAðF4RðbÞÞ\F and jbjXjgjþp: &

Lemma 2.35. There exist a ﬁlter FATðL; AÞ and words b; gARðAÞ such that n gAðF,RðbÞÞ \ðF,RðbÞÞ and eðbÞ¼eðgÞ:

Proof. Let b; g; and F be as in the conclusion of Lemma 2.34. Furthermore, by possibly replacing g by some g0Lg; we may assume without loss of generality that n gAMin ðF4RðbÞÞ: Hence, by Lemma 2.18, gAðF,RðbÞÞ ; thus, by Lemma 2.21(ii), there exists an inference I of g from F,RðbÞ: If all free words of I belongto F; then, by Lemma 2.21(i) and since F is an L-ﬁlter, gAF; a contradiction. Hence, there exists a free word b0 of I that belongs to RðbÞ\F: By Lemma 2.31, there exists an I-inference chain from 26 K.V. Adaricheva et al. / Advances in Mathematics 173 (2003) 1–49 b0 to g; say,

0 (i) b ¼ g0;0 and g ¼ gn;0; (ii) IAg ¼ f nðg ; y; g Þ; for all j n: jþ1;0 kj j;0 j;kj o

It follows from (ii) that jgj;0j¼jgjþ1;0jþ1 for all jon:

Next, we deﬁne inductively b0 ¼ b and biþ1 ¼ bi; for all iop: There are indices 0 0 0 sorpp such that eðbrÞ¼eðbsÞ: Since brKextbLb ; there exists g Kextb such that 0 0 0 brLdilg ; we choose g with minimal length. We observe that jg jXjbrj¼jbjÀ 0 0 0 rXjbjÀpXjgj; hence (since b extends both g and g ) g ¼ gm;0 for some mAf1; y; ng: In particular, g0 is an inner word of I; so

n g0AðF,RðbÞÞ \ðF,RðbÞÞ:

0 n 0 Furthermore, bsLb; thus RðbÞDRðbsÞ; hence g AðF,RðbsÞÞ : Suppose that bsLg : 0 Since brKextbs; there exists dKextg such that brLdild; which contradicts the minimality of g0: Hence we have proved that

0 n\ g AðF,RðbsÞÞ ðF,RðbsÞÞ:

0 Observe that eðg Þ¼eðbrÞ¼eðbsÞ: &

From now on, until the end of the proof of Theorem 2.9, we fix b; g; and F that satisfy the conclusion of Lemma 2.35, and we put u ¼ eðbÞ¼eðgÞ: We observe that geRðbÞ: For a subset G of RðAÞ; we shall say that an inference I from G,RðbÞ is normal,ifbLd implies that bLdild; for any dArng I: We observe that the concept of a normal inference is defined relatively to the (frozen) element b but also to the subset G: For U; GDRðAÞ; we define a subset EðU; GÞ of A and an element EþðU; GÞ of L by

EðU; GÞ¼feðaÞ : aAU-ðG\RðbÞÞg;

_ EþðU; GÞ¼ EðU; GÞ:

The main lemma towards the proof of Theorem 2.9 is then the following:

Lemma 2.36. Let GDRðAÞ; let I beS a normal inference of g from G,RðbÞ: If sAo and y e l; l1; ; ls are words of I with l 1pips rng IðliÞ; then the following inequality holds in L: ! [ þ eðlÞpeðl1Þþ? þ eðlsÞþE rng In rng IðliÞ; G : ð2:7Þ 1pips K.V. Adaricheva et al. / Advances in Mathematics 173 (2003) 1–49 27

Proof. We begin with the following claim.

Claim 1. Let GDRðAÞ; let I be a normal inference of g from G,RðbÞ: Then the inequality

eðlÞpu þ Eþðrng I; GÞð2:8Þ holds for any lArng I:

Proof. If l is a free word of I; then either lAG\RðbÞ in which case eðlÞpEþðrng I; GÞ; or lARðbÞ; in which case, by the normality assumption, bLdill; so eðlÞ¼u: The general case follows by a straightforward induction on jIðlÞj: &

The goal of our next claim is to remove u from inequality (2.8).

Claim 2. Let GDRðAÞ; let I be a normal inference of g from G,RðbÞ: Then the inequality

upEþðrng I; GÞð2:9Þ holds.

Proof. If g is a free word of I; that is, I ¼ /gS; then gAEðrng I; GÞ; so the statement is evident. Now suppose that g is an inner word of I: Hence, there are nonnegative integers kX1 and spk with elements g0; y; gk of rng I such that g0; y; gsÀ1 are inner words, gs; y; gk are free words, and A n I g ¼ fk ðg0; y; gkÞ: a \ Since u ¼ eðgÞ eðgiÞ for all iAf0; y; kg; we get that gs; y; gk belongto G RðbÞ: Hence, the followinginequality holds:

þ eðgÞpeðg0Þþ? þ eðgsÀ1ÞþE ðrng I; GÞ: ð2:10Þ

In view of Claim 1, (2.10), and the join-semidistributive law, we get

þ þ u þ E ðrng I; GÞ¼eðg0Þþ? þ eðgsÀ1ÞþE ðrng I; GÞ

þ ¼ðu4eðg0ÞÞ þ ? þðu4eðgsÀ1ÞÞ þ E ðrng I; GÞ:

Since u is an atom, we get that upEþðrng I; GÞ: &

Now we can conclude the proof of Lemma 2.36. By Corollary 2.26, we can assume 0 without loss of generality that Iðl1Þ; y; IðlsÞ are pairwise disjoint. Put I ¼ 0 0 I½l1; y; ls and G ¼ G,fl1; y; lsg: By Lemma 2.29, I is an (obviously normal) 28 K.V. Adaricheva et al. / Advances in Mathematics 173 (2003) 1–49 inference of g from G0: Then lArng I 0; thus, by Claims 1 and 2,

þ 0 0 eðlÞpE ðrng I ; G Þpeðl1Þþ? þ eðlsÞ ! [ þ þ E rng In rng IðliÞ; G ; 1pips which concludes the proof. &

Finally, we shall get a contradiction that proves that the assumption made at the beginning of the proof of Theorem 2.9 is wrong. This assumption has lead us, see Lemma 2.35, to the existence of FATðL; AÞ and n b; gARðAÞ such that eðbÞ¼eðgÞ and gAðF,RðbÞÞ \ðF,RðbÞÞ: By Lemma 2.21, there exists an inference I of g from F,RðbÞ: By Lemma 2.33, every word of RðbÞ-rng I dilutes b; that is, I is a normal inference from F,RðbÞ: We denote by U the set of inner words of I; and we deﬁne ﬁnite subsets Wn; for nAo; by induction on n; as follows:

W0 ¼fgg:

_ Wnþ1 ¼ Wn,fk /eðaÞS : kAWn and aAU

such that a%Lk and aL/ kg S A for all n o: Then we put W ¼ nAo Wn: We observe that all words of W extend g:

Lemma 2.37. (i) For any kAW; there exists aAU such that eðkÞ¼eðaÞ: (ii) W is ﬁnite. (iii) The relation aLk holds, for any aAU and any Lext-maximal element k of W.

Proof. Statement (i) is obvious by the definition of W: (ii) We claim that there is no xARðAÞ such that jxj4jUj and g_xAW: Suppose _ otherwise, put l ¼jxj and put kn ¼ g ðxjnÞ for all nAf0; y; lg; where xjn denotes the restriction of x to its first n entries. Observe that knAW for all nAf0; y; lg: Furthermore, if nol; then there exists anAU such that anLkn; anL/ kn; and knþ1 ¼ _/ S kn eðanÞ : Since l4jUj; there are m; n in f0; y; l À 1g such that mon and am ¼ _/ S an: From amLkm follows that an ¼ amLkm eðamÞ ¼ kmþ1Lextkn; which contra- jUj dicts the fact that anL/ kn: This proves our claim. Therefore, jWjpp ; so W is finite. (iii) Since both a and k extend g; there exists a largest, with respect to Lext; word n such that nLexta and nLk: Hence, it suffices to prove that n ¼ a: Suppose otherwise, and let d be the successor of n in the Lext-interval from n to a; that is, nLextdLexta and d% ¼ n: It follows from Corollary 2.32 that dAU: Moreover, n ¼ d%Lk while dL/ k by the maximality of n: Hence k_/eðdÞS belongs to W; which contradicts the maximality of k: & K.V. Adaricheva et al. / Advances in Mathematics 173 (2003) 1–49 29

We shall get a contradiction proving that all words from W belongto the ﬁlter F: Indeed, by assumption, geF:

(1) Every maximal element of W belongs to F. Let k be a maximal element of W: For any aAF-rng I; since there exists, by Lemma 2.31, an I-inference chain from a to g and since aag; a% is an inner word of I; thus, by Lemma 2.37(iii), a%Lk: If eðkÞ¼eðaÞ then aLk; whence kAF: So suppose that eðkÞaeðaÞ; for all aAF-rng I: In particular, k_/eðaÞSARðAÞ: Observe that F-rng I is nonempty, and write F-rng I ¼ fa0; y; asg: By Lemma 2.37(i), there exists lAU such that eðkÞ¼eðlÞ: Therefore, by Lemma 2.36

þ eðkÞ¼eðlÞpE ðrng I; FÞpeða0Þþ? þ eðasÞ;

whence sX1 (because eðkÞaeðaiÞ for all i)and

_ _ k ¼ fsðk /eða0ÞS; y; k /eðasÞSÞ: ð2:11Þ

_ _ For all iAf0; y; kg; aiLk; thus aiLk /eðaiÞS; whence k /eðaiÞSAF: So it follows from (2.11) that kAF: (2) Let kag in W such that kKk0 implies that k0AF; for any k0AW: We prove that kAF: By the deﬁnition of W;

k ¼ k% _/eðaÞS for some aAU such that

a%Lk% and aL/ k%: ð2:12Þ

In particular, aLk: Since k is not maximal in W; there exists an inner word d of % I such that dL/ k and dLk: Let d0; y; dt list all such inner words. If lAU such % % that lL/ k; then lag; thus l is an inner word of I; if lLk; then l is one of the di; otherwise, we may start again the process at l% and argue by induction. As a consequence of this,

Every inner word l of I such that

lL/ k extends one of the di: ð2:13Þ

_ For all iAf0; y; tg; k /eðdiÞSAW (because diLk and diL/ k), thus, by the _ induction hypothesis, k /eðdiÞSAF: As diL/ a (otherwise, diLk; a contradiction), a does not belongto rng Iðd0Þ,?,rngS IðdtÞ (we use Lemma 2.22(i)). \ Hence, by Lemma 2.36, if we put X ¼ rng I 0pipt rng IðdiÞ;

þ eðkÞ¼eðaÞpeðd0Þþ? þ eðdtÞþE ðX; FÞ: ð2:14Þ % % Let lAF-X: Then l is an inner word of I; and, by Lemma 2.27, lArng IðlÞ: % Suppose that diLextl; with iAf0; y; tg: Then, by Lemma 2.27, % rng IðlÞDrng IðdiÞ; whence lArng IðdiÞ; a contradiction. It follows then from 30 K.V. Adaricheva et al. / Advances in Mathematics 173 (2003) 1–49

% (2.13) that lLk: Hence, if eðlÞ¼eðkÞ for some lAF-X; then lLk; so kAF: So suppose that eðlÞaeðkÞ for all lAF-X: Write X ¼fl1; y; lug: It follows from (2.14) that

_ _ k ¼ ftþuðk /eðd0ÞS; y; k /eðdtÞS;

_ _ k /eðl1ÞS; y; k /eðluÞSÞ;

whence kAF: By (1) and (2) above, W\fgg is contained in F: (3) Finally, g is an inner word of I; so

_ _ _ _ g ¼ fmÀ1ðg /a1S; y; g /atS; g /b1S; y; g /bmÀtSÞð2:15Þ

_ _ for some mX2 and inner words g /a1S; y; g /atS and free words _ _ _ g /b1S; y; g /bmÀtSAF: As g /aiS belongs to W; it belongs to F by item (2) above. Therefore, it follows from (2.15) that gAF:

This contradiction concludes the proof of Theorem 2.9. &

3. Atomistic and strong (nonnecessarilyﬁnite) convex geometries

3.1. Atomistic convex geometries

As it was already noted in the introduction, finite convex geometries, from the lattice point of view, are dual to antimatroids. In particular, this name indicates that amongcombinatorial objects, they are antithetical to the well-studied objects called matroids. The latter are an abstraction of the idea of linear dependence of vectors in vector spaces. Like finite convex geometries, they can be defined in terms of a closure operator actingon a finite underlyingset (see for example [67]). Specifically, a matroid is a finite zero-closure space ðX; FÞ satisfyingthe exchange property:

xAFðA,fygÞ and xeA imply that yAFðA,fxgÞ;

for all xay in X and all closed ADX: ðEPÞ

A combinatorial geometry (see [37]) is defined as an algebraic, atomistic closure space ðX; FÞ (see Section 1.2) with F satisfyingthe exchangeproperty. The closure lattice of such a space is a geometric lattice, that is, it is algebraic, atomistic, and semimodular. Many examples of geometric lattices appear in connection with various geometrical objects. The subspace lattices of vector spaces as well as the lattices of linear subspaces of projective geometries give the examples of modular geometric lattices. They have been playing a significant role in the development of theory of modular lattices. On the other hand, there exist nonmodular geometric K.V. Adaricheva et al. / Advances in Mathematics 173 (2003) 1–49 31 lattices such as the lattice of affine subspaces of a vector space or the lattice of equivalence relations on a set. The latter were used for special representation of modular and Arguesian lattices [45] (we shall discuss this in more detail in Section 3.8). Not so much was done for extendingthe notion of a finite convex geometry to the infinite case. In [10] the lattices satisfyingthe anti-exchangeproperty (or AEP-lattices), see Section 1.2, were introduced. Like combinatorial geometries, they can also be defined in terms of closure spaces, see Definition 1.6. In our study of infinite convex geometries, we will be focused mostly on atomistic convex geometries. We have already seen in Theorem 1.11 that any finite convex geometry can be ‘‘extended’’ to a finite atomistic convex geometry that is defined on the same underlyingset. Furthermore, all the interestingclasses of closure spaces that we shall mention in the rest of Section 3 are atomistic. Finally, this choice will keep us also closer to the classical notion of a combinatorial geometry. Theorem 1.9 has the followinganalogue for the infinite, atomistic case:

Proposition 3.1. A lattice L is the closure lattice ClðX; FÞ of some atomistic convex geometry ðX; FÞ iff L is complete, atomistic, and satisﬁes SD3ðAt LÞ (see Deﬁnition 1.1), where At L denotes the set of all atoms of L. In particular,ClðX; FÞ is lower semimodular.

Proof. The ﬁrst assertion is evident. To prove the second one, observe that if agb in L; then, by the anti-exchange property, the sets of atoms under a and b differ by a single atom. Then for arbitrary cAL the correspondingsets of atoms under c4a and c4b differ at most by a single atom. This implies lower semimodularity. &

Observe that there are nonjoin-semidistributive atomistic convex geometries. On the other hand, for a given atomistic convex geometry G; let SðGÞ denote the join- subsemilattice of G generated by the atoms of G: If SðGÞ is a sublattice of G; then, by Corollary 1.3, SðGÞ is join-semidistributive. The followingassertion shows that in contrast with infinite combinatorial geometries that may be of finite height, atomistic convex geometries are either finite or ‘‘infinite dimensional’’.

Proposition 3.2. (i) Any join-semidistributive lattice without infinite chains is finite. (ii) Any atomistic convex geometry without infinite chains is finite.

Note. Claim (i) was proved in [47], see also [27, Theorem 5.59]. Claim (ii) generalizes Theorem 8 from [9] since the requirement of biatomicity is superﬂuous there. Its proof follows from (i) and Lemma 1.2. Now we turn to examples to show the diversity of atomistic convex geometries of different nature. Like combinatorial geometries, they emerge in different branches of mathematics. 32 K.V. Adaricheva et al. / Advances in Mathematics 173 (2003) 1–49

3.2. Euclidean spaces

Let En denote the n-dimensional Euclidean space over the ﬁeld of real numbers. A subset XDEn is called convex if X contains the segment ½u; v whenever the points u; v belongto X: The convex hull of a subset YDEn is the smallest convex subset of En containing Y:

Proposition 3.3. The closure space ðEn; CoÞ; where Co denotes the closure operator of taking the convex hull, is an atomistic convex geometry.

The closure lattice CoðEnÞ of this space has been intensively studied in many papers by Bennett (see, for example, [7,8]). This lattice has a large modular core, generated by the afﬁne subspaces of En; see [59]. At the same time it contains a large join-semidistributive sublattice, as indicated by the following Theorem 3.4. Recall that a convex body is a convex compact subset of En: Evidently, all convex bodies with En added as a top element form a complete lattice CBðEnÞ that is a sublattice as well as a complete meet-subsemilattice in CoðEnÞ:

Theorem 3.4. The lattice CBðEnÞ is join-semidistributive.

As a matter of fact, CBðEnÞ is the closure lattice of the atomistic convex geometry ðEn; CBÞ; where CBðYÞ denotes the closure of the convex hull of YDEn if Y is bounded, and En otherwise. The problem of the description of the lattices CBðEnÞ and their sublattices is posed in [38]. Note. For the lattice KðEnÞ of polytopes, that is, ﬁnitely generated convex sets, of En; that is a sublattice of CBðEnÞ; an analogous result was proved in Birkhoff and Bennett [10].

Proof. Any convex body is the closure of the convex hull of the set of its extreme points (see, for example, [22]). Recall that pAX is an extreme point of a convex set X \ if pe½p1; p2 for any p1; p2AX fpg: S Suppose that V ¼ X3Y ¼ X3Z in CBðEnÞ: As X3Y ¼ f½x; y : xAX; yAYg; any extreme point of V lies either in X or in Y: At the same time it should be in X or in Z; hence it lies either in X or in Y-Z: This implies that V ¼ X3ðY-ZÞ: &

On the other hand, CBðEnÞ fails some of the basic properties of algebraic lattices recorded in Lemma 2.1:

Proposition 3.5. For any positive integer n, CBðEnÞ is neither weakly atomic nor dually spatial. K.V. Adaricheva et al. / Advances in Mathematics 173 (2003) 1–49 33

Proof. Let |aCCD be convex bodies and let dAD\C: Consider the segment ½c; d with cAC: As C is closed, one can ﬁnd d1A½c; d; d1ad; with d1AD\C: Then the convex hull F of C,fd1g is, since deF; a convex body such that CCFCD: As for any convex body C in En and xeC one can ﬁnd a larger convex body C0 with xeC0; any convex body is an intersection of larger convex bodies. Thus, CBðEnÞ has no completely meet-irreducible elements. & 3.3. Posets and graphs

For a quasiordered set ðP; pÞ (i.e., p is a reﬂexive and transitive binary relation on P), a subset X of P is (order-) convex,if½x; y¼ftAP : xptpyg is contained in X whenever x; yAX such that xpy: Then the map Co that with every subset X of P associates its convex hull, that is, the least convex subset of P that contains X; is an algebraic zero-closure operator on P: We denote by CoðPÞ¼CoðP; pÞ its closure space. If P% denotes the quotient of P by the equivalence relation deﬁned by x y iff xpypx; then CoðPÞ and CoðP%Þ are naturally isomorphic (see [10, Theorem 6]). Furthermore, in that case, the closure map Co is atomistic. The followingresult, proved in [10, Theorems 6 and 14], implies immediately that if p is antisymmetric, then ðP; CoÞ is an atomistic convex geometry:

Theorem 3.6. The lattice CoðP; pÞ is join-semidistributive.

Theorem 3.6 makes it possible to construct join-semidistributive atomistic convex geometries from other types of combinatorial objects. A directed graph is, by definition, a pair ðV; EÞ where V is a (not necessarily finite) set (of ‘vertices’) and EDV Â V is a set (of ‘edges’). A A A A path from a V to b V is a finite sequence ðviÞ0pipn of vertices with ðvi; viþ1Þ E for 0pion: We shall call a subgraph XDV convex if X contains all paths between any two of its points. Then the map Cgthat with every subset of V associates the least convex subgraph of P that contains X is an algebraic zero-closure operator on V: We denote by CgðV; EÞ the associated closure lattice. Let pE be the reflexive and transitive closure of E: Then it is obvious that CgðV; EÞ¼CoðV; pEÞ: Therefore, by usingTheorem 3.6, we obtain immediately the followingresult:

Theorem 3.7. The lattice CgðV; EÞ of all convex subsets of a graph ðV; EÞ is join- semidistributive.

Note. The easy consequence of Theorem 3.7 that CgðVÞ is lower semimodular was proved for ﬁnite V in [58].

3.4. Lattices of suborders

Consider a pair ðP; LÞ where P is a set and LDP Â P is a strict partial order on P: Let O be the operator of transitive closure deﬁned on BðLÞ: Then the 34 K.V. Adaricheva et al. / Advances in Mathematics 173 (2003) 1–49 closed elements of this closure operator are exactly the partial orders on P contained in L:

Theorem 3.8. The operator O satisﬁes the anti-exchange property, so ðL; OÞ is an algebraic, atomistic convex geometry. Its closure lattice OðLÞ is join-semidistributive iff ðP; LÞ does not contain any inﬁnite bounded chain.

The last statement of Theorem 3.8 is proved in [62]. As a consequence of Theorem 3.8, we get the result, proved in [14], that OðLÞ is lower semimodular.

Proposition 3.9. If a poset P contains an inﬁnite bounded chain, then OðPÞ is not lower continuous.

Proof. Suppose, for example, that P contains a strictly increasingchain of the form /an :0pnpoS: We put A ¼fðai; ajÞ : iojoog and Bn ¼fðai; ajÞ : npiojpog; forT all nAo: Then ða0; aoÞAA3Bn for all nAo; while e ða0; aoÞ A ¼ A3 nAo Bn: &

3.5. Semilattices

Let ðL; 4Þ be a meet-semilattice. One can consider three different anti-exchange closures on L: For XDL; we put

Sub4ðXÞ¼subsemilattice of L generated by X;

( subsemilattice of L generated by X if X is finite; Subf ðXÞ¼ L otherwise;

SubcðXÞ¼complete subsemilattice of L generated by X

ðin case L is completeÞ:

In the above deﬁnitions, it is convenient to include | amongthe (complete) subsemilattices of L:

Theorem 3.10. Let L be a meet-semilattice (resp. a complete meet-semilattice). Then the closure spaces ðL; Sub4Þ and ðL; Subf Þ (resp., ðL; SubcÞ) are atomistic convex geometries.

We shall denote by Sub4ðLÞ; Subf ðLÞ; and SubcðLÞ; respectively, the correspond- ingclosure spaces. The necessary and sufﬁcient conditions for these lattices to be join-semidistributive were found in [1]. In particular, K.V. Adaricheva et al. / Advances in Mathematics 173 (2003) 1–49 35

Theorem 3.11. (1) For arbitrary ðL; 4Þ; the lattice Subf ðLÞ is join-semidistributive. (2) If L does not contain inﬁnite increasing chains, then both Sub4ðLÞ and SubcðLÞ are join-semidistributive.

We also observe the followinganalogueof Proposition 3.5:

Proposition 3.12. Let L be a meet-semilattice (resp. a complete meet-semilattice). Then Subf ðLÞ (resp., SubcðLÞ) is weakly atomic but not necessarily dually spatial.

Proof. Let SCT be finite (complete) subsemilattices of L and let tAT\S: If T1 is the finite (complete) subsemilattice of L generated by S,ftg; then S1 ¼ T1\ftg is a finite (complete) subsemilattice of L: Hence, SDS1!T1DT: Denote by Rþ the complete meet-semilattice of positive real numbers Rþ with þ þ natural ordering. Any XASubcðR Þ; XaR ; is disjoint from some interval ½r1; r2 0 0 0 0 with r1or2: Pick elements r1; r2 such that r1or1or2or2: Then the equality X ¼ 0 0 ðX,fr1gÞ-ðX,fr2gÞ shows that X is meet-reducible in Sub4ðLÞ: The same holds þ for XASubf ðR Þ: &

As a consequence, neither Subf ðLÞ nor SubcðLÞ is necessarily algebraic (see Lemma 2.1). The followingexamples show that the situation with dual algebraicity is not better.

Example 3.13. Let L be an inﬁnite chain. Then any subset of L is a subsemilattice. It follows that L is the only dually compact element of Subf ðLÞ: In particular, Subf ðLÞ is not dually algebraic, while it is, of course, lower continuous.

The next example is taken from the description of dually algebraic lattices Sub4ðLÞ in [1].

Example 3.14. Let L ¼ðo þ 1Þ,ftg; where the ordinal o þ 1 is endowed with its natural orderingand t is a new element such that 0otoo and t is incomparable with all positive integers. So L is a complete meet-semilattice. Then neither Sub4ðLÞ nor SubcðLÞ is dually algebraic (it is not even lower continuous).

3.6. Universal Horn logic and lattices of algebraic subsets

We recall that LqðKÞ denotes the lattice of all subquasivarieties of a given quasivariety K: If LqðKÞ is atomistic then it is an atomistic convex geometry since any quasivariety lattice is join-semidistributive. But the study of such lattices has brought a wider class of atomistic convex geometries. We have already mentioned the fact, proved in [35], that for any algebraic lattice A; the lattice SpðAÞ is isomorphic to LqðKÞ for some quasivariety K of relational structures. 36 K.V. Adaricheva et al. / Advances in Mathematics 173 (2003) 1–49

For arbitrary complete lattices, we can state the followingresult that was kindly offered to us by F. Wehrung:

Proposition 3.15. Let L be a complete lattice. Then SpðLÞ satisﬁes SD3ðAt LÞ (see Deﬁnition 1.1). In particular, SpðLÞ is the closure lattice of an atomistic convex geometry.

¼ ¼ % Proof. We put L ¼ L\f1g; and, for a subset X of L ; we define SpðXÞ¼X\f1g; % where X denotes the algebraic subset of L generated by X: Then Sp is an atomistic closure operator on BðL¼Þ; and the rule X/X,f1g defines an isomorphism from ¼ ClðL ; SpÞ onto SpðLÞ: ¼ To conclude the proof, it suffices to verify that ðL ; SpÞ satisfies (AEP). Let X be a closed subset of L¼: It is convenient to observe first the followingpreliminary claim. For any cAL¼; the subset X,½0; c is a closed subset of L¼ and it contains X,fcg; ¼ whence SpðX,fcgÞDX,½0; c: Now let aab be elements of L \X: Suppose that SpðX,fagÞ ¼ SpðX,fbgÞ: By the claim above, aAX,½0; b and bAX,½0; a; hence, since a; beX; we obtain that a ¼ b; a contradiction. &

Remark 3.16. It follows from this and from Lemma 1.2 that the join-semidistributive law

X3Y ¼ X3Z ) X3Y ¼ X3ðY-ZÞ holds in SpðLÞ for arbitrary X and ﬁnite Y; Z: Of course, for L complete and upper continuous, a stronger statement holds, namely, SpðLÞ is join-semidistributive, see Theorem 2.2(i).

Our next result gives, for a complete upper continuous lattice A; a convenient description of the completely meet-irreducible elements of SpðAÞ:

Lemma 3.17. Let A be a complete, upper continuous lattice. We put

½a-b¼fxAA : apx ) bpxg; for all a; bAA:

Then the completely meet-irreducible elements of SpðAÞ are exactly the subsets of the form ½a-b; where a!b in A and a is compact. Furthermore, the upper cover of ½a-b in SpðAÞ is ½a-b,fag:

Note. By Theorem 2.2(i), SpðAÞ is lower continuous, hence all its completely meet- irreducible elements are dually compact.

Proof. The veriﬁcation of the fact that ½a-b is a completely meet-irreducible - element of SpðAÞ; with upper cover ½a b,fag; is straightforward.V Conversely, let P be a completely meet-irreducible element of SpðAÞ: We put a ¼ ðA\PÞ: For all xAA; we denote by x% the least element of P above x: K.V. Adaricheva et al. / Advances in Mathematics 173 (2003) 1–49 37

Claim 1. The element a belongs to A\P; and it is compact. Furthermore, P,fag is the (unique) upper cover of P in SpðAÞ:

Proof. Suppose that aAP: Since P is completelyT meet-irreducible, in order to reach a contradiction, it is sufﬁcient to prove that P ¼ uAA\P P½u; where we deﬁne P½u as the algebraic subset of A generated by P,fug: We observe that

P½u¼P,fu4x : xAPg¼P,fxAA : x ¼ u4x%g: T \ % A \ % If x is an element of uAA\P P½u P; then x ¼ x4u for all u A P; whence x ¼ x4a: But from x%AP and aAP follows that xAP; a contradiction. Hence, aeP; while xAP for all xoa: It follows immediately that P,fag belongs to SpðAÞ; from which the last statement of the claim follows. Finally, let I an up-directedW poset and let ðaiÞiAI be an increasingfamily of A elements of A such that ap iAI ai: Suppose that a4/ ai for all i I; thatW is, a4aioa: A A Hence a4ai P; whence, by usingthe upper continuity of A; a ¼ a4 iAI ai P; a contradiction. Therefore, a is compact. &

Claim 2. The equality a ¼ a%4x holds, for all xAA\P:

Proof. From PCP½x and from Claim 1 follows that aAP½x\P; that is, a ¼ a%4x: &

Claim 3. P ¼½a-a%:

Proof. The containment PD½a-a% follows immediately from the deﬁnition of a%: Conversely, let xA½a-a%; suppose that xeP: From the deﬁnition of a follows that apx; thus a%px by assumption, thus a ¼ a% by Claim 2, a contradiction. &

Suppose that there exists xAA such that aoxoa%: From Claim 3 follows that xeP; then from Claim 2 follows that a ¼ x; a contradiction. Hence, a!a%; which concludes the proof. &

As an immediate corollary, even for an algebraic lattice A; the lattice SpðAÞ does not need to be dually spatial, even though, according to Theorem 2.2(ii), it is dually algebraic:

Corollary3.18. Let A be the ideal lattice of a dense chain with zero. Then SpðAÞ has no completely meet-irreducible elements.

Proposition 3.19. Let X be a set. Then the closure lattice SpðBðXÞÞ is dually spatial.

Note. Observe that if A is an arbitrary algebraic lattice and X is the set of its compact elements, then SpðAÞ is a principal ideal in SpðBðXÞÞ; see [35]. 38 K.V. Adaricheva et al. / Advances in Mathematics 173 (2003) 1–49

Proof. It follows from Lemma 3.17 that the completely meet-irreducible elements of SpðBðXÞÞ are the DðC; tÞ¼½C-C,ftg; for a ﬁnite subset C of X and an element t of X\C: For arbitrary ZASpðBðXÞÞ; we denote by ZC the least element of Z above C; for every ﬁnite subset C of X: Then it is easy to verify that \ Z ¼ fDðC; tÞ : CDX finite; tAZC\Cg; whence Z is a meet of completely meet-irreducible elements of SpðBðXÞÞ: &

It is interestingto study whether a givenlattice of the form SpðAÞ can be decomposed as a direct product of directly indecomposable factors. In Wehrung[66], it is proved that for any complete Boolean lattice B; the lattice SpðBÞ is subdirectly irreducible (thus directly indecomposable). On the other hand, if A is deﬁned as the ideal lattice of the real unit interval ½0; 1; then SpðAÞ cannot be decomposed as a direct product of directly indecomposable factors, see [5]. Some more information about direct decompositions can be found in Proposition 3.20.

3.7. Deﬁning strong convex geometries

Table 1 gives a list of the principal classes of atomistic convex geometries mentioned in this paper. All the correspondinglattices L are atomistic, complete, and satisfy the weak join-semidistributive law SD3ðAt LÞ: The list below provides some additional explanation for the entries of Table 1.

(1) See Proposition 3.5. (2) For A the ideal lattice of any dense chain with zero, SpðAÞ is not even dually spatial, see Corollary 3.18. (3) See Proposition 3.12.

Table 1 Lattices and closure spaces

n n Lattice CoðE Þ CBðE Þ CoðPÞ OðLÞ SpðAÞ Sub4ðLÞ Subf ðLÞ SubcðLÞ n n \ Closure space ðE ; CoÞðE ; CBÞðP; CoÞðL; OÞðA f1g; SpÞðL; Sub4ÞðL; Subf ÞðL; SubcÞ

Algebraic Yes Noð1Þ Yes Yes Noð2Þ Yes Noð3Þ Noð3Þ Dually algebraic Noð4Þ Noð5Þ Noð4Þ Noð6Þ Yes (A algebraic) Noð8Þ Noð9Þ Noð8Þ Noð7Þ (general case) Weakly atomic Yes Noð1Þ Yes Yes Yes Yes Yes Yes ð10Þ ð11Þ ð13Þ ð13Þ SD3 No Yes Yes No Yes (A upper No Yes No continuous) Noð12Þ (general case) K.V. Adaricheva et al. / Advances in Mathematics 173 (2003) 1–49 39

(4) The lattice is not lower continuous. Take P ¼ o; put TX ¼f0g and Yn ¼½n; NÞ for all nAo: Then X3Yn ¼ Y0 for all n; while X3 Yn ¼ X: nAo (5) The top element is the only dually compact one. (6) OðLÞ is not lower continuous as a rule, see Proposition 3.9. (7) See Example 2.3. (8) Neither Sub4ðLÞ nor SubcðLÞ need to be lower continuous, see Example 3.14. (9) The lattice Subf ðLÞ is lower continuous but it need not be dually algebraic, see Example 3.13. (10) For nX2; see, for example, [10, p. 234]. (11) See Theorem 3.8 and [62]. (12) If A is an upper continuous complete lattice, then SpðAÞ is join-semidistributive, see Theorem 2.2(i). For non upper continuous A; this property may fail, see Theorem 2.8. (13) See Theorem 3.11 and [1].

It seems that the notion of atomistic convex geometry is too general to achieve a satisfactory theory. On the other hand, the additional requirement for a closure operator to be algebraic as in combinatorial geometry would imply the loss of some important examples. For instance, the lattice SpðAÞ of algebraic subsets of an algebraic lattice A is not algebraic while it is dually algebraic. We believe that the general deﬁnition of a strong convex geometry should be well balanced in the followingsense:

(1) it should include a rich list of examples, in particular amongthose listed in Table 1; (2) it should be restrictive enough to imply reasonably strong structural results.

In particular, the importance of the example of SpðAÞ makes it impossible to deﬁne a strong convex geometry to be algebraic, in contrast with the case of combinatorial geometries. On the other hand, we observe that the properties to be dually spatial and weakly atomic are shared by algebraic lattices (see Lemma 2.1) as well as the special types of SpðAÞ considered in Proposition 3.19. It turns out that complete atomistic dually spatial lattices inherit the decomposition properties of algebraic lattices. For example, the following statement is a consequence of a result in [66], thus extendingLibkin’s decomposition Theorem [51] to the dually spatial case:

Proposition 3.20. Any complete atomistic dually spatial lattice can be decomposed as a direct product of directly indecomposable lattices.

Amongthe results that one could hope for in (2) would be, for example, results of spatial theory of join-semidistributive lattices, that would be, ideally, counterparts of the modular case established in [41]. In view of Example 3.25, this is probably not trivial to achieve. 40 K.V. Adaricheva et al. / Advances in Mathematics 173 (2003) 1–49

On the other hand, the atomistic weakly atomic convex geometries carry the feature of finite convex geometry expressed in property (4) of Theorem 1.7, which is often considered as the definition of a convex geometry in combinatorics. Basically, weak atomicity says in this context that there should exist plenty of closed subsets that can be extended by a single point to another closed subset. We also believe that lower semimodularity should be a sound property in strongconvex geometry, so it requires an abundance of covers in the lattice of closed sets. Thus we propose the followingdefinition of a strongconvex geometry:

Deﬁnition 3.21. A closure space ðX; FÞ is (i) almost algebraic, if its closure lattice is weakly atomic and dually spatial (see the statement of Lemma 2.1). (ii) a strong convex geometry, if it is an almost algebraic atomistic convex geometry.

In particular, the following‘‘implications’’ follow rightaway from Lemma 2.1:

algebraic atomistic convex geometry

) strongconvex geometry

) atomistic convex geometry:

We observe that Table 1 provides numerous examples that show that the implications above cannot be reversed.

3.8. Embedding theorems

In [29] it was proved that any complemented modular lattice can be embedded into a projective geometry. The latter may be viewed as the (modular) lattice of closed sets of the underlyingprojective space. This cannot be done for arbitrary modular lattices, see, for example, [18] for a counterexample. Later on, the idea of ordered projective spaces whose subspace lattices are modular and spatial was developed, see [24]. It allowed to prove the following embeddingresult, see [23].

Theorem 3.22. Any modular lattice can be embedded into an algebraic spatial modular lattice.

Our Theorem 2.4 together with observations about the lattices SpðAÞ and SpðBðXÞÞ made in the previous subsection can be viewed as an analogous result applied to join-semidistributive lattices and atomistic convex geometries. K.V. Adaricheva et al. / Advances in Mathematics 173 (2003) 1–49 41

Theorem 3.23. Any ﬁnitely presented join-semidistributive lattice can be embedded into a dually algebraic, dually spatial, join-semidistributive, atomistic convex geometry.

Proof. By Theorem 2.4, we can embed any ﬁnitely presented join-semidistributive lattice into SpðAÞ for some algebraic lattice A: Furthermore, SpðAÞpSpðBðXÞÞ where X is the set of compact elements of A: It is proved in Theorem 2.2 that SpðBðXÞÞ is both join-semidistributive and dually algebraic. By Proposition 3.19, SpðBðXÞÞ is dually spatial. &

The followingexample shows that Theorem 3.23 cannot be extended from ﬁnitely presented to ﬁnitely generated join-semidistributive lattices:

Example 3.24. There exists a four-generated join-semidistributive lattice that cannot be embedded into any join-semidistributive lower continuous (in particular, dually algebraic) lattice. Let us consider the lattice L4 depicted on the left half of Fig. 2 (see also [44, Fig. 4.6(i)]). We observe that L4 is join-semidistributive. LetV M be a join- % % semidistributive lowerV continuousV lattice extending L4: Put b ¼ iX1 bi: If b ¼ b; then d ¼ d3b ¼ d3ð iX1 biÞ¼ iX1 ðd3biÞ¼b1V; a contradiction. % % % % % Thus, we should have b4b: Then b4/ a: Set c ¼ iX1 ci: Clearly, b4c ¼ 0: Let us % prove that a3c% ¼ a3b: Indeed, cipa3bi for all iX1; hence ^ ^ % c%p ða3biÞ¼a3 bi ¼ a3b: iX1 iX1

The proof of the converse inequality is similar. Finally, it follows from the join- % % % semidistributivity of M that a3b ¼ a3ðb4c%Þ¼a; hence bpa; a contradiction.

The followingexample, communicated to us by F. Wehrung,satisﬁes an even stronger incompleteness property but it is not ﬁnitely generated.

Example 3.25. There exists a join-semidistributive lattice that cannot be embedded into any complete join-semidistributive lattice.

Proof. Let E be the lattice diagrammed on the right half of Fig. 2. Then E is join- semidistributive. Suppose that E embeds into a join-semidistributive lattice L in which every bounded countable sequence admits a supremum. We put _ _ a ¼ an; b ¼ bn: nAo nAo

From the fact that anobn3c and bnoanþ13c for all nAo follows that a3c ¼ b3c; hence, since L is join-semidistributive, apða4bÞ3c: However, apu; bpv; and u4v ¼ 0; hence a1oapðu4vÞ3c ¼ c; a contradiction. & 42 K.V. Adaricheva et al. / Advances in Mathematics 173 (2003) 1–49

a0 v u a1 b1 a2 c1 a3

b2 a4 c a5 2

d b3

b2 a a2 L4 b1 c b c a1

a0 b0 0 E 0 Fig. 2.

What can be said about embeddingarbitrary join-semidistributive lattices? Recall that a lattice L is called biatomic (see [9]) if it is atomic, that is, any element contains an atom, and for any atom xAL and u; vAL; if xpu3v then xpp3q for some atoms ppu and qpv: This property plays an essential role in the theory of combinatorial geometries, for, as was proved in [9], the closure lattice of a combinatorial geometry is biatomic iff it is modular. It seems that biatomicity is much less restrictive for convex geometries. It is amazing that the atomistic convex geometries in all but one of the examples given in Sections 3.2–3.6 are biatomic! The only exception is the lattice OðLÞ of orders contained in a given order ðX; LÞ: So it is not surprisingthat biatomic atomistic convex geometries give room for arbitrary join semidistributive lattices.

Theorem 3.26. Any join-semidistributive lattice can be embedded into an atomistic, algebraic, biatomic convex geometry.

Proof. By Corollary 2.7, the quasivariety of join-semidistributive lattice is generated by the lattices SpðAÞ with algebraic A: These lattices are biatomic. This can be either K.V. Adaricheva et al. / Advances in Mathematics 173 (2003) 1–49 43 proved directly or deduced from the fact that they are isomorphic to some lattices of quasivarieties which are always biatomic [4]. Observe that the property of an element of a lattice to be an atom is a ﬁrst-order property. Hence, the biatomicity is a ﬁrst order property. Further, let x be the followingsentence:

x"8x; y ðxoy ) (zððz is an atomÞ & ðzpyÞ & ðz4/ xÞÞ:

Then a lattice L with zero satisﬁes x iff L is atomistic. Observe that L may not be complete. Now let L be an arbitrary join-semidistributive lattice. By Theorems 1.4 and 2.4, LASPufSpðAÞ : A algebraic latticeg: It follows that L can be embedded into an atomistic (generally noncomplete), biatomic, and join-semidistributive lattice L1: Let A be the set of atoms of L1; and let BiðAÞ be the lattice of all subsets Y of A such that cpa3b and a; bAY implies that cAY; for all a; b; cAA: Then BiðAÞ is an algebraic subset of the powerset of A; let X/BiðXÞ be the associated closure operator on A: It is obviously an algebraic closure operator. It follows easily from the biatomicity of L1 that the join of elements in BiðAÞ has the form:

X þBi Y ¼fzAA : (ðx; yÞAX Â Y; zpx3yg;

for all X; YABiðAÞ:

This implies immediately that BiðAÞ is biatomic. A a A \ A Let X BiðAÞ; a b in A; a BiðX,fbgÞ X: Then apL1 x3b for some x X: A 0 0A 0 Assume that b BiðX,fagÞ; then bpL1 x 3a for some x X: We get ðx3x Þ3a ¼ 0 0 ðx3x Þ3b ¼ c; which implies that c ¼ x3x since L1 is join-semidistributive, whence a; bpx3x0: Together with XABiðAÞ; this implies that aAX; a contradiction. Hence, Bi is an anti-exchange closure operator. It remains to prove that L1 embeds into BiðAÞ: Deﬁne a map f from L1 to BðAÞ by the rule fðxÞ¼faAA : apxg; for all xAL1: Clearly, fðxÞABiðAÞ for any xAL1: As L1 is atomistic, f is injective. Moreover, f preserves all existingmeets. Now let x; yAL1: Then, by usingthe biatomicity of L1;

fðx3yÞ¼faAA : apx3yg

¼faAA : (u; vAA; upx; vpy; and apu3vg

¼ fðxÞþBi fðyÞ; for all x; yAL1; which concludes the proof. &

The embeddingof an arbitrary atomistic biatomic and join-semidistributive lattice into an atomistic, algebraic, biatomic convex geometry given in the previous proof does not preserve the join-semidistributive law in general. For example, by this construction, SpðAÞ is embedded into Sub4ðAÞ which is generally not join- semidistributive, see [1]. Another example gives the lattice CBðEnÞ of convex bodies 44 K.V. Adaricheva et al. / Advances in Mathematics 173 (2003) 1–49 of En as well as its noncomplete sublattice KðEnÞ of convex polytopes, because they are both embedded in the same algebraic lattice CoðEnÞ of convex subsets of En where the join-semidistributive law fails. Because of Example 3.25, one cannot hope to be able to embed an arbitrary join-semidistributive lattice into a complete join- semidistributive lattice, let alone a join-semidistributive convex geometry. Still, the followingproblem remains open:

Problem 2. Can every ﬁnitely generated join-semidistributive lattice be embedded into some complete join-semidistributive lattice?

The join-semidistributive lattice of Example 3.25 cannot be embedded into any complete join-semidistributive lattice, but it is not finitely generated. Let us see now what is known about lattices which are embeddable into members of given classes of finite atomistic convex geometries. By Theorem 1.11, any finite join-semidistributive lattice can be embedded into a finite atomistic convex geometry. This result is the best possible one because finite convex geometries are join-semidistributive as well. However, it would be interestingto find out which join- semidistributive lattices can be embedded into some specific finite convex geometries. In particular, given a class C of finite join-semidistributive lattices, what are the lattices that can be embedded into some member of C? One such subclass is well known. Recall [54] that a finitely generated lattice L is lower bounded, if there exists a homomorphism g : FLðnÞ7L from a finitely generated free lattice onto L such that any element xAL has a least pre-image. In [6], the definition was extended, callinga lattice A lower bounded, if every finitely generated sublattice of A is lower bounded. It is known that lower bounded lattices inherit the join-semidistributive law from free lattices and form a proper subclass in SD3; see [27] for the most comprehensive account. Not all finite convex geometries are lower bounded. For example the lattice CoðPÞ of convex subsets of a finite poset P is not lower bounded unless P does not contain any four-element chain. Some other types of convex geometries are not only lower bounded but contain any finite lower bounded lattice as a sublattice, which can be seen in the next theorem.

Theorem 3.27. Let L be a ﬁnite lattice. Then the following are equivalent:

(i) L is lower bounded; (ii) L can be embedded into Sub4ðAÞ for some ﬁnite (semi)lattice A; (iii) L can be embedded into OðLÞ for some order ðX; LÞ on a ﬁnite set X.

Proof. Implications ðiÞ3ðiiÞ follow from results in [3,61], while ðiÞ3ðiiiÞ follow from results in [63]. &

Observe that it was proved by Dilworth in the 1940s [16] that any ﬁnite lattice is embedded into the closure lattice of some ﬁnite combinatorial geometry. Four decades later, Pudla´ kandTuma’ [60] strengthened this result by proving that any K.V. Adaricheva et al. / Advances in Mathematics 173 (2003) 1–49 45

finite lattice is isomorphic to a sublattice of the lattice EqðXÞ of equivalence relations on a finite set X: It follows from Theorem 3.27 that finite Sub4ðAÞ and OðLÞ cannot serve as ‘‘universal’’ convex geometries, that is, those containing all finite join- semidistributive lattices as sublattices. Not much is known about the sublattices of the lattices CoðPÞ; for a finite poset P; or CoðEn; XÞ; for a finite set X of points in En:

Problem 3. Is there a special type of finite convex geometry that contains all finite join-semidistributive lattices as sublattices? In particular, describe the subclasses of finite join-semidistributive lattices embeddable into finite lattices of the form CoðPÞ or CoðEn; XÞ:

Notice that the representation problem was solved for some finite convex geometries of the above forms. The finite lattices representable as Sub4ðAÞ; for some finite semilattice A; were characterized in [2]. In [10], the description of lattices of the form CoðPÞ is given for any poset P: In this regard it is worthwhile to mention the problem of Jamison about which convex geometries are representable as CoðE2; XÞ for some X: Some approaches to the solution can be found in [68]. One could also ask the followingrelated question:

Problem 4. Can any ﬁnite join-semidistributive lattice be embedded into some ﬁnite biatomic atomistic convex geometry?

Theorems 2.8, 2.4, and 3.27 provide now an interestinganalogywith Jo ´ nsson’s three-type embeddings into lattices EqðXÞ of equivalence relations. More precisely, Jo´ nsson proved [45] that (i) any lattice has type-3 embeddinginto some EqðXÞ; (ii) the lattices with type-2 embeddingcompose the class of all modular lattices, and (iii) the lattices havingtype-1 embeddingare Arguesian. Our theorems give a classification of lattices embeddable into SpðAÞ with respect to the properties of complete lattice A: (i) any lattice is embeddable into SpðAÞ with dually algebraic A; (ii) any finitely presented join-semidistributive lattice is a sublattice of SpðAÞ with A being both algebraic and dually algebraic, and (iii) the sublattices of SpðAÞ with finite A compose a class of finite lower bounded lattices. In view of Theorem 2.2, this classification is not complete yet:

Problem 5. Which join-semidistributive lattices can be embedded into SpðAÞ for some complete and upper continuous lattice A?

As proved in [33], for any upper continuous lattice A; the lattice SpðAÞ is lower continuous. Thus, Example 3.25 shows that the lattices embeddable into SpðAÞ with upper continuous A form a proper subclass of join-semidistributive lattices. 46 K.V. Adaricheva et al. / Advances in Mathematics 173 (2003) 1–49

To strengthen the similarity between the two classiﬁcations above, one could observe that the class of lower bounded lattices plays in some sense an analogous role in SD3 as Arguesian lattices do in the class M of modular lattices (see more details in [28]). Note that as for combinatorial geometries, we can claim that every lattice can be embedded into some atomistic convex geometry. Indeed, by Theorem 2.8, every lattice can be embedded into a lattice of the form SpðTÞ for some dually algebraic lattice T; while, by Proposition 3.15, SpðTÞ is the closure lattice of a convex geometry. In fact, every lattice can be embedded into even a strong convex geometry, as the followingresult by Bredikhin and Schein [11] shows:

Theorem 3.28. Any lattice L can be embedded into the algebraic, atomistic convex geometry OðLÞ of all orders containing a given order ðX; LÞ; with jXj¼ maxf@0; jLjg:

Acknowledgments

While observations on the topic have been collected by the authors in a course of several years, the structural core of the paper was written duringfirst two authors’s stay at Technische Hochschule Darmstadt in springof 1997, as a part of a joint Russian–German research project. We appreciate the interest of our German colleagues and the encouragement of their seminar where the results of this paper were presented for the first time. Many corrections improving the presentation of results were suggested by post- graduate students of Novosibirsk State University Michail Sheremet, Marina Semenova, Vladimir Hudyakov and Alexandr Kravchenko, who did the proof- readingof the early versions of the paper. When the paper was in progress, Marina has been keepingin touch, continuingto send us her helpful comments. The new stage of preparation of the paper started when Prof. F. Wehrung undertook the challenging task of deep proofreading and editing the paper. As the result of his endless efforts not only the proofs of the core results were considerably amplified but the whole structure of the paper underwent a serious modification, includingthe major definitions, statements of the problems and the set of key examples. His contribution to this work cannot be overestimated. Many thanks go also to R. Wehrung for his artistic advice on better picture presentation. The first author is grateful to Prof. R. McKenzie for the invitation to present the results of this paper on the seminar at Vanderbilt University, in February of 2001. The results about finite convex geometries were discussed also on the logical seminar at University of Wisconsin in Madison, in May of 2001, duringthe visit of the first author arranged by Prof. S. Lempp. We are thankful to Prof. J. Tuma,’ Prof. M. Wild and Prof. M. Haviar for their comments and interest in the topic. Several useful comments and corrections were suggested by anonymous referee. K.V. Adaricheva et al. / Advances in Mathematics 173 (2003) 1–49 47

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