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Infinite combinatorial issues raised by lifting problems in universal algebra

The critical point between varieties A and B of algebras is defined as the least cardinality of the semilattice of compact congruences of a member of A but of no member of B, if it exists. The study of critical points gives rise to a whole array of problems, often involving lifting problems of either diagrams or objects, with respect to functors. These, in turn, involve problems that belong to infinite combinatorics. We survey some of the combinatorial problems and results thus encountered. The corresponding problematic is articulated around the notion of a k-ladder (for proving that a critical point is large), large free set theorems and the classical notation (k,r,l){\to}m (for proving that a critical point is small). In the middle, we find l-lifters of posets and the relation (k,


Introduction
The main goal of the present paper is to give a rough survey of some combinatorial, often set-theoretical issues raised by recent work on representation problems in universal algebra. It is an expanded version of the author's talk at the BLAST 2010 conference in Boulder, Colorado.
Let us first quickly describe the universal algebraic background. A signature (or similarity type) consists of a set Σ, together with a map ar : Σ → {0, 1, 2, . . . } (the arity map). An algebra is a structure A = (A, (f A | f ∈ Σ)), where A is a nonempty set and for each f ∈ Σ, the "fundamental operation" f A is a map from A ar(f ) to A, identified with the unique constant in its range in case ar(f ) = 0. A congruence of A is an equivalence relation on A which is compatible with all the fundamental operations of A. The set Con A of all congruences of A forms an algebraic lattice under set inclusion, and we denote by Con c A its semilattice of compact elements. Hence Con c A is the (∨, 0)-semilattice of all finitely generated congruences of A. It is a central question in universal algebra to determine which algebraic lattices can be represented as Con A, for an algebra A in a given class of algebras. As it turns out that infinite (∨, 0)-semilattices are often more conveniently dealt with than infinite algebraic lattices, the class of questions above can be rephrased as which (∨, 0)-semilattices can be represented as Con c A, for an algebra A in a given class of algebras. A variety of algebras is the class of all the algebras satisfying a given set of identities (in a given signature).
Many combinatorial features of the work under discussion in the present paper are contained in the notion of critical point between varieties of algebras. The following definition was introduced in the survey paper Tůma and Wehrung [42], then slightly amended in Gillibert [10,11]. For a class C of algebras, we set Con c C := {S | (∃A ∈ C)(S ∼ = Con c A)} , the compact congruence class of C.
Definition 1.1. The critical point crit(A; B), between classes A and B of algebras, is the least possible cardinality of a member of (Con c A) \ (Con c B) if Con c A ⊆ Con c B, and ∞ if Con c A ⊆ Con c B).
Thus the critical point measures the "containment defect" of Con c A into Con c B. It turns out that estimating the critical point between varieties of algebras, in particular varieties of lattices, is a highly nontrivial task, of which we shall give a rough outline in Section 7. Proving that a critical point is large requires tools called ladders; we present a foretaste of the methods used in Sections 3 (for 1ladders) and 4 (for 2-ladders). Although no universal-algebraic use of 3-ladders has been found so far (in my optimistic moments I do hope that there will be some), these objects raise fascinating set-theoretical questions. Furthermore, the failure of three-dimensional amalgamation is, in some sense, at the basis of nontrivial counterexamples (cf. Růžička, Tůma, and Wehrung [40] or Wehrung [47]), and one cannot completely discard the possibility of future uses of that property, see Section 5.
On the other hand, proving that a critical point is small involves quite different combinatorial tools, articulated around the notion of a λ-lifter (Section 6) and the more amenable relation (κ, <λ) ; P , for infinite cardinals κ and λ and a poset P (cf. Section 9). It turns out that the relation (κ, <λ) ; P , when specialized to truncated cubes P , gives a relation known to infinite combinatorists, namely the (κ, r, λ) → m relation (cf. Notation 10.1 and Proposition 10.2). The more algebraic viewpoint brought by the (κ, <λ) ; P framework made it possible, in Gillibert and Wehrung [16], to establish new relations of the form (κ, r, λ) → m, of which we give an overview in Section 10.
In the middle of "large critical point results" (involving ladders) or "small critical point results" (involving the relation (κ, <λ) ; P ), there is a work of category theory (cf. Gillibert and Wehrung [17]), introducing the categorical notion of a larder and the combinatorial notion of a lifter (cf. Section 6). These tools are mainly designed to introduce, and use, the Condensate Lifting Lemma CLL, and its main precursor the Armature Lemma. These results make it possible to turn diagram counterexamples to object counterexamples. While there are surprisingly many statements that can be proved that way (beginning with those involving critical points), we present in Section 8 a quite isolated diagram counterexample, due to Tůma and the author, for which there is still no known associated object counterexample.

Basic concepts
2.1. Set theory. We shall use basic set-theoretical notation and terminology about ordinals and cardinals. We denote by f "X the image of a set X under a function f . Cardinals are initial ordinals. We denote by cf α the cofinality of an ordinal α. We denote by ω := {0, 1, 2, . . . } the first limit ordinal, mostly denoted by ℵ 0 in case it is viewed as a cardinal. We identify every non-negative integer n with the finite set {0, 1, . . . , n−1} (so 0 = ∅). We denote by κ + the successor cardinal of a cardinal κ, and we define κ +n , for a non-negative integer n, by κ +0 := κ and κ +(n+1) = (κ +n ) + .
We denote by P(X) the powerset of a set X, and we set for every cardinal κ.
By "countable" we shall always mean "at most countable".

2.2.
Partially ordered sets (posets). All our posets will be nonempty. For posets P and Q, a map f : We denote by 0 P the least element of P if it exists. An element p in a poset P is join-irreducible if p = X implies that p ∈ X, for every (possibly empty) finite subset X of P ; we denote by J(P ) the set of all join-irreducible elements of P , endowed with the induced partial ordering. We set for all subsets Q and X of P ; in case X = {a} is a singleton, then we shall write Q ↓ a instead of Q ↓ {a}, and so on. We say that P is lower finite if P ↓ a is finite for each a ∈ P . A subset Q of P is a lower subset of P if P ↓ Q = Q. We say that P is a tree if P has a smallest element and P ↓ a is a chain for each a ∈ P . An ideal of a poset P is a nonempty, upward directed, lower subset of P .
We recall the definitions of a few classical numerical invariants associated to a finite poset P : -The width of P , denoted by wd(P ), is the maximal cardinality of a pairwise incomparable subset of P . -The breadth of P , denoted by br(P ), is the least natural number m such that for all x i , y i ∈ P , for 0 ≤ i ≤ m, if x i ≤ y j for all i = j, then there exists i such that x i ≤ y i . -The order-dimension of P , denoted by dim(P ), is the least possible natural number m such that P embeds, as a poset, into a product of m chains. [19,Section II.5]).
3. An example in universal algebra: representing countable distributive semilattices Let us start with an example. We are given a class C of algebras, and we assume that every finite Boolean lattice is isomorphic to Con c A for some A ∈ C. Can we extend this representation result to countable distributive (∨, 0)-semilattices?
Without additional assumptions, this cannot be done. For example, let C := D be the variety of all distributive lattices. Then every finite Boolean lattice B belongs to Con c D (for Con c B ∼ = B), but for any D ∈ D, the (∨, 0)-semilattice Con c D is generalized Boolean, thus it is a lattice; and many distributive (∨, 0)-semilattices (even countable ones) are not lattices.
Can we find reasonable assumptions, on the class C, that make it possible to represent any countable distributive (∨, 0)-semilattice, starting with a finite representability result?
Here is the required finite representability result.
One-dimensional amalgamation property for Con c . For each A ∈ C, each finite Boolean semilattice S, and each (∨, 0)-homomorphism ϕ : Con c A → S, there are B ∈ C, a homomorphism f : A → B, and an isomorphism ε : Con c B → S such that ϕ = ε • Con c f .
The statement above involves a yet undefined notation, namely Con c f , where f is no longer an object, but a morphism. Nor surprisingly, Con c f is defined as the map from Con c A to Con c B that to every compact congruence α of A associates the congruence of B generated by {(f (x), f (y)) | (x, y) ∈ α}. The map Con c f is a (∨, 0)-homomorphism from Con c A to Con c B. We obtain the missing piece of information that is necessary in order to understand the "amalgamation" statement above, with a plus about direct limits (called, in category theory, directed colimits). The one-dimensional amalgamation property is illustrated in Figure 3.1.
Starting from this. . . Now suppose that the one-dimensional amalgamation property holds for the functor Con c on the class C. Let S be a countable (∨, 0)-semilattice. Using a result from Bulman-Fleming and McDowell [3] (see also Goodearl and Wehrung [18] for a self-contained proof), there is a representation of the form where all the S n are finite Boolean (∨, 0)-semilattices and all the transition maps of the direct system in (3.1) are (∨, 0)-homomorphisms. Suppose that S n ∼ = Con c A n , with an isomorphism ε n : Con c A n → S n , and denote by σ n : S n → S n+1 the transition map associated with the direct system in (3.1). We get a (∨, 0)-homomorphism σ n • ε n : Con c A n → S n+1 , thus, by the one-dimensional amalgamation assumption, there are A n+1 ∈ C, a homomorphism f n : A n → A n+1 , and an isomorphism ε n+1 : Con c A n+1 → S n+1 such that ε n+1 •Con c f n = σ n •ε n . Defining A as the direct limit of the system we obtain, as the functor Con c preserves direct limits, that Con c A ∼ = S. So we have obtained the following result.
Proposition 3.2. Let C be a class of algebras of the same signature. If the onedimensional amalgamation property holds for Con c on C, then every countable distributive (∨, 0)-semilattice is isomorphic to Con c A for the direct limit A of some countable sequence in C.
This approach is used in Růžička, Tůma, and Wehrung [40] to establish the following statement. The corresponding statement with "countable" replaced by "with at most ℵ 1 elements" is still open. Yet there are also known positive results in cardinality ℵ 1 , for which we shall set the framework in the next section. It will turn out that none of the corresponding representation results holds at cardinality ℵ 2 . 4. One step further: representing semilattices of cardinality ℵ 1 In the context of Section 3, one may try to represent larger distributive (∨, 0)semilattices than just the countable ones. Thus the first step consists of trying semilattices of cardinality ℵ 1 . The naïve approach, consisting of keeping the onedimensional amalgamation property and replacing the chain ω of all natural numbers by the larger chain ω 1 of all countable ordinals, obviously fails: if card S = ℵ 1 , then there is no way to represent S = lim − →ξ<ω1 S ξ with all S ξ finite. The relevant strengthening of the assumption on C is the following.
Two-dimensional amalgamation property for Con c . Let A 0 , A 1 , A 2 be members of C with homomorphisms f i : A 0 → A i for i ∈ {1, 2}, and let S be a finite Boolean (∨, 0)-semilattice with (∨, 0)-homomorphisms ψ i : Con c A i → S, for i ∈ {1, 2}, such that ψ 1 • Con c f 1 = ψ 2 • Con c f 2 . Then there are a member A of C, homomorphisms g i : A i → A for i ∈ {1, 2}, and an isomorphism ε : The two-dimensional amalgamation property can be illustrated by On that picture, the plain vanilla arrows are those that are assumed to exist from the start, while the dotted ones are those whose existence follows from the property.
Assuming two-dimensional congruence amalgamation, we still need to replace the chain ω by a poset of cardinality ℵ 1 that makes it possible to use two-dimensional congruence amalgamation. We remind the reader that lower finite posets are defined in Section 2.2.
Definition 4.1. Let k be a positive integer. A lower finite lattice L is a k-ladder if every element of L has at most k lower covers in L.
Con c f 1 e e u u u u u u u u u  Note that every k-ladder has breadth at most k. The 1-ladders are exactly the finite chains and the chain ω of all natural numbers. Note that k-ladders are called k-frames in H. Dobbertin [6]. Due to a terminology conflict with another latticetheoretical concept, the von Neumann frames, we chose in Grätzer, Lakser, and Wehrung [21] to call these objects ladders. The concept originates in S. Z. Ditor [5], where k-ladders are simply called k-lattices.
Ditor proves in [5] that every k-ladder has at most ℵ k−1 elements. (This is also a straightforward consequence of Kuratowski's Free Set Theorem, see Kuratowski [32].) More relevant to our present congruence representation problem, Ditor proves the following. Now suppose that two-dimensional congruence amalgamation holds. We prove that every distributive (∨, 0)-semilattice S with at most ℵ 1 elements is isomorphic to Con c A, for some direct limit A of members of C. Write S = lim − → S, for a directed poset P with at most ℵ 1 elements and a direct system S = (S p , σ q p | p ≤ q in P ) of finite Boolean (∨, 0)-semilattices and (∨, 0)-homomorphisms. For example, P can be assumed to be the set of all finite subsets of S × ω, partially ordered under set inclusion. Let F be a 2-ladder of cardinality ℵ 1 . As F is lower finite, it is easy to construct, by induction, an isotone cofinal map ν : F → P ; this reduces the situation to the case where P = F . Then, using two-dimensional amalgamation and the definition of a 2-ladder, one constructs, by induction within the ladder, an As the functor Con c preserves direct limits, Con c A ∼ = S. We have thus outlined the proof of the following. Proposition 4.3. Let C be a class of algebras of the same signature. If the twodimensional amalgamation property holds for Con c on C, then every distributive (∨, 0)-semilattice with at most ℵ 1 elements is isomorphic to Con c A for the direct limit A of a diagram, indexed by a 2-ladder of cardinality ℵ 1 , of members of C.
This argument has turned out to be very useful in proving various representation results, in size ℵ 1 , with respect to the Con c functor, starting with A. P. Huhn [24,25]; see, for example, Grätzer, Lakser, and Wehrung [21], or Růžička, Tůma, and Wehrung [40]. See also Wehrung [46] for a representation result with respect to another functor than Con c , namely the compact ideal semilattice functor on a regular ring.

Looking ahead: 3-dimensional amalgamation and 3-ladders
For higher cardinalities than ℵ 1 , the situation gets somehow romantic. There are, to this date, no known congruence representation results specifically stated in cardinality ℵ 2 . Yet there are theorems, now of set-theoretical nature, which are an order of magnitude harder than those about representation in cardinality below ℵ 1 . These results go together with an array of still unsolved problems.
We first deal with the easy problem of 3-dimensional congruence amalgamation, for a class C of algebras with the same signature. Set B 3 := P(3) (recall that 3 = {0, 1, 2}), partially ordered under set inclusion, and set B = 3 := B 3 \ {3}. The statement of three-dimensional amalgamation is quite discouraging at first sight: 3 ) be a natural transformation from Con c A to the one-object diagram S. Then there are A ∈ C and a natural transformation 3 . This can be explained as follows. We are given a truncated cube (here, a B = 3indexed diagram) A of members of C, and we form the image Con c A of this diagram under the Con c functor. At the top of this diagram, we put the finite Boolean semilattice S, thus getting a cube S of (∨, 0)-semilattices. Three-dimensional amalgamation states that there is a way to adjoin a top vertex to A, thus getting a cube B, such that Con c B ∼ = S, in a way that preserves the already existing relations. This is illustrated in Figure 5.1, with α p := Con c α p for each p ∈ {{0, 1}, {0, 2}, {1, 2}}. The unlabeled arrows should be obvious: for example, the arrow from {0} ; and so on. Unfortunately, the following result shows that three-dimensional amalgamation occurs only in exceptional cases.
Proposition 5.1. Suppose that a class C of algebras of the same signature satisfies three-dimensional amalgamation.
Proof. Assume that f ↾ A0 = g↾ A0 . Set 2 := {0, 1}, and for each i ∈ {1, 2}, denote by σ i : Con c A i → 2 the unique (∨, 0)-homomorphism that sends every nonzero element to 1. We apply three-dimensional amalgamation to the diagrams represented in Figure 5.2, minus A, Con c A, and the dotted arrows. Note that the one-to-oneness of both f and g ensures that both maps f := Con c f and g := Con c g separate zero, thus the right hand side diagram commutes. Three-dimensional amalgamation ensures the existence of A together with homomorphisms u, v : A 1 → A and w : A 2 → A with an isomorphism ε : Con c A → 2 such that both diagrams of Figure 5.2 (dotted arrows included) commute. Again, we are using the notational convention w := Con c w, and so on. The unlabeled arrows should be obvious, for example the arrow from A 0 to A 1 is the inclusion map.  From the commutativity of the right hand side diagram it follows that σ 2 = ε•w. As σ 2 separates zero and ε is an isomorphism, w separates zero, which means that w is one-to-one. From the commutativity of the left hand side diagram it follows that As w is one-to-one, we obtain that f = g.
Proposition 5.1 suggests at first sight that it is quite hopeless to achieve any representation proof in cardinality ℵ 2 via three-dimensional amalgamation. Although some hope remains to apply three-dimensional amalgamation on special configurations, such as strong amalgamation, no such application has been found so far.
Even in case, some day, someone manages to work out a usable version of threedimensional amalgamation, another catch is the existence of ladders. Indeed, in order to be able to extend the proof of Proposition 4.3 to classes satisfying some version of three-dimensional amalgamation, we need the existence of a 3-ladder of cardinality ℵ 2 . This problem was first raised in Ditor [5].
The following result, by Wehrung [48], shows that a positive solution of Ditor's Problem is at least consistent with ZFC. In particular, as every L[A], for A ⊆ ω 1 , has a gap-1 morass (cf. Devlin [4]), it follows that If there is no 3-ladder of cardinality ℵ 2 , then ω 2 is inaccessible in the constructible universe L.
The axiom MA(ℵ 1 ; precaliber ℵ 1 ), which is Martin's Axiom restricted to collections of ℵ 1 dense subsets in posets of precaliber ℵ 1 (cf. Weiss [51, Section 3]), holds in a generic extension of the universe by a suitable notion of forcing satisfying the countable chain condition (cf. Jech [26,Section 16]). On the other hand, a gap-1 morass exists in a generic extension by a suitable countably closed notion of forcing (cf. Velleman [44,Section 3.3]; see also Brooke-Taylor and Friedman [2, Proposition 25]). As countably closed posets with the countable chain condition are, from the forcing viewpoint, trivial, the two notions of forcing involved in those generic extensions cannot be the same: they must be, in some sense, "orthogonal"! This suggests a positive solution to Ditor's Problem in ZFC, nevertheless no such solution has been found so far.
Our proof of Theorem 5.2 uses the equivalent form of morasses called simplified morasses introduced in Velleman [45].
It seems reasonable to expect that higher morasses, or higher versions of Martin's Axiom, would imply the existence of k-ladders of cardinality ℵ k−1 for k ≥ 4. In an ideal world, Ditor's Problem would first be settled in ZFC alone. However, we are not living in an ideal world. . .

CLL, larders, and lifters
Propositions 3.2 and 4.3 are typical illustrations of a "finite" representation result on the functor Con c making it possible to prove an "infinite" representation result. Now there are situations where the problematic is just opposite: namely, going from an infinite representation result to a finite one. Finite would appear as a limit case of infinite! A paradigm for such situations (critical points) will shortly be given in Section 7.
The arguments underlying Propositions 3.2 and 4.3 are best set in a category theory context. Basically, we are given categories A, B, S together with functors Φ : A → S and Ψ : B → S. We are trying to find an assignment Γ : A → B such that Φ(A) ∼ = ΨΓ(A), naturally in A, on a "large" subcategory of A. We can paraphrase this further by saying that we are trying to make Γ "as functorial as possible". Hence we need an assumption of the form "for many A ∈ A, there exists B ∈ B such that Φ(A) ∼ = Ψ(B)". The situation is illustrated in Figure 6.1.
This problem is tackled, in Gillibert and Wehrung [17], by the introduction of a whole theory, involving a categorical concept called a larder.  S ⇒ ⊆ S. These add-ons are required to satisfy a few obvious-looking preservation properties (e.g., directed colimits), plus a less obvious-looking "Löwenheim-Skolem type" property with respect to the functor Ψ. An important additional attribute of a larder, reminiscent of the First Isomorphism Theorem in group theory, is called projectability. Most applications of our theory involve a result called the "Condensate Lifting Lemma", or CLL, which we paraphrase below.
Condensate Lifting Lemma. Suppose that A, B, S, Φ, and Ψ are part of a projectable larder and let P be a "nice" poset. If In this statement, A P denotes the category whose objects are all the direct systems (A p , α q p | p ≤ q in P ), with A p an object of A and α q p : A p → A q , with the natural transformations as morphisms. We shall go back to the meaning of "nice poset" in a moment.
To paraphrase CLL further, if Ψ lifts many objects, then it lifts many (P -indexed) diagrams. Yet another way to see this is that CLL turns diagram counterexamples to object counterexamples. Applications of CLL, or its main precursor the Armature Lemma, include the following.
-Let V be a nondistributive variety of lattices. Then the free lattice (resp., free bounded lattice) in V on ℵ 1 generators has no congruence-permutable, congruence-preserving extension (cf. Gillibert and Wehrung [17]). -There exists a non-coordinatizable sectionally complemented modular lattice (without unit), of cardinality ℵ 1 , with a large 4-frame (cf. Wehrung [49]). -There exists a lattice of cardinality ℵ 1 , in the variety generated by the five-element modular non-distributive lattice M 3 , without any congruence n-permutable, congruence-preserving extension for any positive integer n (cf. Gillibert [13]). -Let A and B be varieties of lattices such that A is contained neither in B nor its dual, and every simple member of B has a prime interval. Then the critical point crit(A; B) (cf. Definition 1.1) lies below ℵ 2 (cf. Gillibert [14]). -There is a unital exchange ring R, of cardinality ℵ 3 , such that the monoid V(R) of all isomorphism types of finitely generated right R-modules is not isomorphic to V(B) for any ring B that is either von Neumann regular or a C*-algebra of real rank zero (cf. Wehrung [50]).
-Let A and B be locally finite varieties of algebras such that every finite (∨, 0)-semilattice has, up to isomorphism, only finitely many liftings, with respect to the functor Con c , in B, and every such lifting is finite. ( [23].) If Con c A is not contained in Con c B, then crit(A; B) ≤ ℵ 2 (cf. Gillibert [15]). There is still one completely undefined term in the statement of CLL above, namely "nice", as an attribute of a poset. Not every poset qualifies as "nice". The definition of "niceness", formally existence of a λ-lifter (λ is a certain infinite cardinal depending of the data), is relatively complex and thus we shall relegate it to Section 11 (cf. Definition 11.2). Nonetheless, it is still possible to describe with relatively little effort which posets are dealt with by CLL, and which are not. Observe that these definitions have, at first sight, little to do with category theory. Notation 6.1. Let X be a subset in a poset P . We denote by ▽X the set of all minimal elements of P ⇑ X. Definition 6.2. We say that a subset X in a poset P is ▽-closed if ▽Y ⊆ X for any finite Y ⊆ X. The ▽-closure of a subset X of a poset P is the least ▽-closed subset of P containing X. We say that P is • a pseudo join-semilattice if for each finite X ⊆ P there exists a finite Y ⊆ P such that P ⇑ X = P ↑ Y (cf. Section 2.2); • supported if P is a pseudo join-semilattice and the ▽-closure of every finite subset of P is finite; • an almost join-semilattice if P is a pseudo join-semilattice and P ↓ a is a join-semilattice for each a ∈ P .
Every (∨, 0)-semilattice (thus, in particular, every lattice with zero) is an almost join-semilattice. Furthermore, every almost join-semilattice is supported and every supported poset is a pseudo join-semilattice. Both converse containments fail, as illustrated in Figure 6.2; the leftmost poset is not a pseudo join-semilattice. Posets that are not almost join-semilattices As mentioned above, "nice" in the statement of CLL means having a λ-lifter, where, roughly speaking, λ is an upper bound on the sizes of the categorical data. This brings a severe restriction on the shape of P , involving a connection between the objects of Definition 6.2 and the definition of a lifter (Definition 11.2): namely, If P has a λ-lifter, then it is an almost join-semilattice. More precisely, the following result is proved in Gillibert and Wehrung [17]. It relates existence of lifters, almost join-semilattices, and the more familiar-looking (at least for those familiar with large free sets problems) infinite combinatorial statement in (ii). Theorem 6.3. Let λ be an infinite cardinal and let (X, X) be a λ-lifter of a poset P . Put κ := card X. Then the following statements hold: (i) P is a disjoint union of finitely many almost join-semilattices with zero.
(ii) For every isotone map F : [κ] <cf(λ) → [κ] <λ , there exists a one-to-one map σ : P κ such that In particular, as none of the posets represented on Figure 6.2 is an almost joinsemilattice, none of them has a lifter. As to the statement of CLL, these posets are not "nice".
Item (ii) of Theorem 6.3 bears an intriguing relationship with the problem of well-foundedness of liftable posets. The following result is proved in Gillibert and Wehrung [17].

Critical points between varieties
A good illustration of the various combinatorial principles involved in Section 6 is provided by the theory of critical points (cf. Definition 1.1). An upper bound for many critical points is provided by the following deep and difficult result, proved in Gillibert [10,11]. This result is extended to quasivarieties (and relative congruence lattices) in Gillibert and Wehrung [17]. Moreover, by using the results of Tame Congruence Theory TCT (cf. Hobby and McKenzie [23]), it is shown in Gillibert and Wehrung [17] that Theorem 7.1 can be extended to the case where B has finite signature, is finitely generated, and omits both TCT types 1 and 5. In particular, the latter condition holds in case B satisfies a nontrivial congruence lattice identity. The strongest refinement of Theorem 7.1 to date, obtained in Gillibert [15], is for A and B both locally finite with B "strongly congruence-proper" (which holds in case B omits both TCT types 1 and 5); furthermore, the conclusion is strengthened there to crit(A; B) ≤ ℵ 2 . The bound ℵ 2 is sharp.
For lattice varieties, Theorem 7.1 is refined further in Gillibert [14]. The proof of Theorem 7.2 is extremely involved, and it uses a precursor of CLL (cf. Section 6) called the Armature Lemma. It is not known whether the assumption about simple members of B can be removed from the assumptions of Theorem 7.2.
Establishing an estimate of the form crit(A; B) > ℵ 0 requires an analogue of Proposition 3.2, in particular a direct limit along the chain ω of all natural numbers. Likewise, establishing an estimate of the form crit(A; B) > ℵ 1 requires an analogue of Proposition 4.3, in particular a direct limit along a 2-ladder of cardinality ℵ 1 . On the other hand, proving an estimate of the form crit(A; B) ≤ ℵ m requires different techniques, the combinatorial aspects of which we shall outline in Section 9. In many proofs of such estimates, one first finds a finite diagram A in A, indexed by a finite lattice P , such that Con c A is never isomorphic to Con c B for any P -indexed diagram B in B; then, using CLL, this diagram counterexample is turned to an object counterexample in A.
Before going back, in Section 9, to the combinatorial aspects of such arguments, we shall see in Section 8 that certain problems may have diagram counterexamples but no known object counterexamples.

A diagram counterexample without any known associated object counterexample
The Congruence Lattice Problem (CLP), raised by R. P. Dilworth in the forties, asked whether every distributive algebraic lattice is isomorphic to the congruence lattice of a lattice; equivalently, whether every distributive (∨, 0)-semilattice is isomorphic to Con c L for a lattice L. This problem got finally settled in the negative in Wehrung [47]; the sharp bound for the size of the counterexample (viz. ℵ 2 ) was obtained in Růžička [39]. For an account of the problem, see Grätzer [20].
While the first version of that negative answer was obtained for congruence lattices of lattices, it got soon extended to a much wider class of structures by using deep results in commutator theory by Kearnes and Szendrei [27]. The result can be stated as follows.
Theorem 8.1. There exists a diagram D ⊲⊳ , indexed by a finite poset, of finite Boolean semilattices and (∨, 0, 1)-embeddings, which cannot be lifted, with respect to the Con c functor, in any variety satisfying a nontrivial congruence lattice identity.
In particular, the diagram D ⊲⊳ cannot be lifted, with respect to the Con c functor, by lattices, majority algebras, groups, loops, modules, and so on. On the other hand, Lampe [33] proved that every (∨, 0, 1)-semilattice (distributive or not) is isomorphic to Con c G for some groupoid G (a groupoid in universal algebra is just a nonempty set with a binary operation). By Gillibert [10,11], this result can be extended to any diagram of (∨, 0, 1)-semilattices and (∨, 0, 1)-homomorphisms indexed by a finite poset, and in particular to D ⊲⊳ : that is, D ⊲⊳ can be lifted by a diagram of groupoids.

P(4) P(3)
u 2 a a g g g g g g g g e a a g g g g g g g g e  turn it into an object counterexample. In particular, it is still unknown whether every distributive (∨, 0)-semilattice is isomorphic to Con c M for some majority algebra M . (A majority algebra is a nonempty set M , endowed with a ternary operation m, such that m(x, x, y) = m(x, y, x) = m(y, x, x) = x for all x, y ∈ M . It is well-known that the congruence lattice of a majority algebra is distributive.) Therefore, the representation problem of distributive (∨, 0)-semilattices as compact congruence semilattices of majority algebras has a diagram counterexample (namely D ⊲⊳ ) but no known object counterexample. It is very difficult to prove that the containment of the congruence class of all lattices into the one of all majority algebras is proper ; this is done in Ploščica [37].

The Kuratowski index of a finite poset
The complexity of the definition of a lifter (Definition 11.2) makes its verification quite unpractical, even for easily described finite posets. The following much more user-friendly variant is introduced in Gillibert and Wehrung [16].
As proved in Gillibert and Wehrung [16], if P is lower finite, then it is sufficient to replace P ↓ z by J(P ) ↓ z in the statement above. This makes the verification of the statement (κ, <λ) ; P much more convenient on given finite posets.
The following result from Gillibert and Wehrung [17] tells exactly when a small enough poset has a lifter (cf. Definition 11.2), and gives an estimate, in terms of the (κ, <λ) ; P notation, of the size of such a lifter.
Theorem 9.2. Let λ and κ be infinite cardinals and let P be a lower finite poset in which every element has less than cf(λ) upper covers. Then the following are equivalent: (i) P has a λ-lifter (X, X) such that X consists of all principal ideals of X and card X = κ, while X is a lower finite almost join-semilattice. (ii) P has a λ-lifter (X, X) such that card X = κ. (iii) P is a finite disjoint union of almost join-semilattices with zero, and (κ, <λ) ; P .
The following definition and result can be found in Gillibert and Wehrung [16].
Definition 9.3. The Kuratowski index of a finite poset P , denoted by kur(P ), is defined as 0 if P is pairwise incomparable, and, otherwise, as the least n > 0 if it exists such that (λ +(n−1) , <λ) ; P for each infinite cardinal λ.
The following corollary can be found in Gillibert and Wehrung [17].
There are finite posets for which the Kuratowski index is easy to calculate, see Gillibert and Wehrung [16].
Proposition 9.6. The following statements hold, for any finite posets P and Q.
(i) kur(P ) = 1 whenever P is a nontrivial finite tree.
In particular, it follows from Proposition 9.6(iv) that kur(2 n ) = n for every natural number n; this is part of Kuratowski's Free Set Theorem (cf. Kuratowski [32]).
However, there are also easily described finite lattices for which the Kuratowski index is unknown. For example, consider the finite lattices P and Q represented in Figure 9.1. Both P and Q have breadth two and order-dimension three. As P P Q We do not know which of those inequalities are actually equalities. A simple infinite combinatorial statement equivalent to kur(P ) = 2 (resp., kur(Q) = 2) is given in Gillibert and Wehrung [16,Section 6].

Large free sets and truncated cubes
Establishing upper bounds for various critical points, due either to Gillibert or to Ploščica, makes a heavy use of large free sets. We recall the following notation, see Erdős et al. [8].
By using Propositions 9.4, 9.6, and 10.2, it is possible to get new estimates for large free sets results. The first such estimate is the following. For any positive integer n, it is not hard to verify that br B n+2 ( n) = dim B n+2 ( n) = n + 1. By Proposition 9.4, it follows that kur B n+2 ( n) = n+1. We obtain the following result of Gillibert [10], see also Gillibert and Wehrung [16] for a self-contained proof.
The large free set relation (ℵ 4 , 4, ℵ 0 ) → 6 is obtained by finding an upper bound (in fact, here, an exact evaluation) of the order-dimension of B 6 ( 4) (namely, 5). Now the problem of estimating the order-dimensions of truncated cubes B m ( r) is a whole topic, started by Dushnik [7] and going on in various works such as Brightwell et al. [1], Füredi and Kahn [9], Kierstead [28,29], Spencer [41]. Using the various estimates available, we obtain in Gillibert and Wehrung [16] new large free sets relations, of which a sample is the following.
(Although it looks quite strange that the size of the free set is one more than a power of 2 in (iii) while it is a power of two in (iv), this seems to be due to the formulas involved having nothing to do with each other.) Proposition 10.6(i,ii) uses Dushnik's exact estimate from [7], while (iii) uses Füredi and Kahn's estimate from [9], and (iv) uses the estimate attributed in Spencer [41] to Hajnal. Among those estimates, the best asymptotic behavior is enjoyed by Hajnal and Spencer's one. This estimate makes it possible to prove that (ℵ n , r, ℵ 0 ) → E(n, r) with lg lg E(n, r) ∼ n r2 r log 2 as n ≫ r ≫ 0 (where lg x denotes the base 2 logarithm while log x denotes the natural logarithm).

Appendix: norm-coverings and lifters
In the present section we shall give formal definitions for some of the objects involved in Section 6. We restrict ourselves to infinite combinatorial objects (normcoverings, lifters) and omit the more complicated categorical objects (larders), for which formal definitions can be found in Gillibert and Wehrung [17]. We remind the reader that pseudo join-semilattices and supported posets are defined in Section 6.
We start with an auxiliary definition.
Definition 11.1. Let P be a poset.
• A norm-covering of P consists of a pseudo join-semilattice X together with an isotone map ∂ : X → P . • An ideal x of X is sharp if {∂x | x ∈ x} has a largest element, then denoted by ∂x. • Then we set X = := {x ∈ X | ∂x is not maximal}, for every set X of sharp ideals of X.

Now we can define lifters.
Definition 11.2. For an infinite cardinal λ, a λ-lifter of a poset P is a pair (X, X), where X is a norm-covering of P , X is a set of sharp ideals of X, and (i) Every principal ideal of X = has cardinality < cf(λ).
12. Open problems 12.1. Ladders. Let us first restate Ditor's Problem [5] about ladders (cf. Section 5). So far, we know that a positive solution to Problem 1 is consistent with ZFC, cf. Theorem 5.2. Even more, every universe of ZFC has a ccc generic extension and a countably closed generic extension both satisfying the existence of a 3-ladder of cardinality ℵ 2 .
In fact, Ditor asks in [5] whether there exists a k-ladder of cardinality ℵ k−1 (which is the upper bound given by Kuratowski's Free Set Theorem), for k ≥ 3. Because of Theorem 5.2, we can also ask the following weaker variant.
Problem 2. Let k ≥ 3 be an integer. Is the existence of a k-ladder of cardinality ℵ k−1 consistent with ZFC?
12.2. Well-foundedness of liftable posets. We ask in Gillibert and Wehrung [17] whether, for any infinite cardinal λ, any poset with a λ-lifter is well-founded. Equivalently, we ask whether (ω + 1) op has no λ-lifter, for any infinite cardinal λ. As explained at the end of Section 6, this is related to the following question. Problem 3. Does there exist an infinite cardinal κ such that for any isotone mapping F : [κ] ℵ0 → [κ] ℵ0 , there exists a sequence (κ n | n < ω) from κ such that κ n / ∈ F ({κ i | i > n}) for each n < ω?
Recall that the isotonicity assumption simply means that x ⊆ y implies that F (x) ⊆ F (y) for any countable subsets x and y of κ. It can be easily seen that if we remove this isotonicity assumption, then the corresponding problem has no solution (cf. the end of the proof of Proposition 3-6.3 in Gillibert and Wehrung [17]). Fred Galvin and Pierre Gillibert observed independently that every cardinal κ solving positively Problem 3 should be greater than or equal to ℵ ω .

Critical points.
Our first problem about critical points asks whether these are absolute (in the set-theoretical sense). A related question is raised in Gillibert and Wehrung [17], namely whether critical points can be calculated by a recursive algorithm. This would of course imply a negative answer to Problem 4. It is plausible that the construction of an example with crit(A; B) = ℵ 3 would involve 3-ladders (cf. Problem 1) together with some form of three-dimensional amalgamation (see Section 5).
12.4. Kuratowski index. We remind the reader that a challenging question, raised in Gillibert and Wehrung [16,Section 6], is to determine the Kuratowski indexes of the two finite lattices P and Q represented in Figure 9.1. Another puzzling question is the following (cf. Notation 10.1). Problem 6. Does (ℵ 4 , 4, ℵ 0 ) → 7?
An even far more ambitious question is to ask whether one can determine the largest integer m such that ZFC proves the relation (ℵ 4 , 4, ℵ 0 ) → m. All we know so far is that 6 ≤ m < t 4 (cf. Section 10). This leaves plenty of room.