Large semilattices of breadth three
A 1984 problem of S. Z. Ditor asks whether there exists a lattice of cardinality , with zero, in which every principal ideal is finite and every element has at most three lower covers. We prove that the existence of such a lattice follows from either one of two axioms that are known to be independent of , namely (1) Martin’s Axiom restricted to collections of dense subsets in posets of precaliber , (2) the existence of a gap- morass. In particular, the existence of such a lattice is consistent with , while the non-existence of such a lattice implies that is inaccessible in the constructible universe.
We also prove that for each regular uncountable cardinal and each positive integer , there exists a -semilattice of cardinality and breadth in which every principal ideal has less than elements.
Key words and phrases:Poset; lattice; breadth; lower cover; lower finite; ladder; Martin’s Axiom; precaliber; gap-1 morass; Kurepa tree; normed lattice; preskeleton; skeleton
2000 Mathematics Subject Classification:Primary 06A07; Secondary 03C55; 03E05; 03E35
Various representation theorems, stating that every object of ‘size’ belongs to the range of a given functor, rely on the existence of lattices called -ladders. By definition, a -ladder is a lattice with zero, in which every principal ideal is finite, and in which every element has at most two lower covers. Every -ladder has cardinality at most , and the existence of -ladders of cardinality exactly was proved in Ditor  (cf. Proposition 4.3). These -ladders have been used in various contexts such as abstract measure theory (Dobbertin ), lattice theory (Grätzer, Lakser, and Wehrung ), ring theory (Wehrung ), or general algebra (Růžička, Tůma, and Wehrung ). A sample result, established in , states that Every distributive algebraic lattice with at most compact elements is isomorphic to the lattice of all normal subgroups of some locally finite group. (Here and in many related results, the bound turns out to be optimal.)
The basic idea of “ladder proofs” is always the same: we are given categories and together with a functor and a ‘large’ object of (of ‘size’ ), that we wish to represent as , for some object in the domain of . We represent as a direct limit of (say) ‘finite’ objects , where is an upward directed poset of cardinality (often the lattice of all finite subsets of in case we are dealing with a concrete category). Then, using the existence of a -ladder of cardinality , we can replace the original poset by that -ladder. Under “amalgamation-type” conditions, this makes it possible to represent each as some , with transition morphisms between the s being constructed in such a way that can be defined and .
We define -ladders the same way as -ladders, except that “two lower covers” is replaced by “three lower covers”. The problem of existence of -ladders of cardinality was posed in Ditor . Such -ladders would presumably be used in trying to represent objects of size . Nevertheless I must reluctantly admit that no potential use of the existence of -ladders of cardinality had been found so far, due to the failure of a certain three-dimensional amalgamation property, of set-theoretical nature, stated in Section 10 in Wehrung , thus making Ditor’s problem quite ‘romantic’ (and thus, somewhat paradoxically therefore arguably, attractive).
However, this situation has been evolving recently. For classes and of algebras, the critical point is defined in Gillibert  as the least possible cardinality of a semilattice in the compact congruence class of but not of if it exists (and, say, otherwise). In case both and are finitely generated lattice varieties, it is proved in Gillibert  that is either finite, or for some natural number , or . In the second case only examples with have been found so far (Ploščica [12, 13], Gillibert ). Investigating the possibility of (i.e., ) would quite likely require -ladders of cardinality .
The present paper is intended as an encouragement in that direction. We partially solve one of Ditor’s problems by giving a (rather easy) proof that for each regular uncountable cardinal and each positive integer , there exists a -semilattice of cardinality and breadth in which every principal ideal has less than elements (cf. Theorem 5.3). Furthermore, although we are still not able to settle whether the existence of a -ladder of cardinality is provable in the usual axiom system of set theory with the Axiom of Choice, we prove that it follows from either one of two quite distinct, and in some sense ‘orthogonal’, set-theoretical axioms, namely a weak form of Martin’s Axiom plus denoted by (cf. Theorem 7.9) and the existence of a gap- morass (cf. Theorem 9.1). In particular, the existence of a -ladder of cardinality is consistent with , while the non-existence of a -ladder of cardinality implies that is inaccessible in the constructible universe.
Our proofs are organized in such a way that no prerequisites in lattice theory and set theory other than the basic ones are necessary to read them. Hence we hope to achieve intelligibility for both lattice-theoretical and set-theoretical communities.
2. Basic concepts
We shall use standard set-theoretical notation and terminology. Throughout the paper, ‘countable’ will mean ‘at most countable’. A cardinal is an initial ordinal, and we denote by the successor cardinal of a cardinal . More generally, we denote by the successor cardinal of , for each natural number . We denote by (resp., ) the domain (resp., range) of a function , and we put for each . We denote by the powerset of a set , and by the partial operation of disjoint union. A function is finite-to-one if the inverse image of any singleton under is finite. For a set and a cardinal , we set
Two elements and in a poset are comparable if either or . We say that is a lower cover of , if and there is no element such that ; in addition, if is the least element of (denoted by if it exists), we say that is an atom of . We say that is atomistic if every element of is a join of atoms of . For a subset and an element in , we set
We say that is a lower subset of , if for each . (In forcing terminology, this means that is open.) An ideal of is a nonempty, upward directed, lower subset of ; it is a principal ideal if it is equal to for some . A filter of is an ideal of the dual poset of . We say that is lower finite if is finite for each . An order-embedding from a poset into a poset is a map such that iff , for all . An order-embedding is a lower embedding if the range of is a lower subset of . Observe that a lower embedding preserves all meets of nonempty subsets in , and all joins of nonempty finite subsets of in case is a join-semilattice.
For an element and a subset in , we denote by the least element of above if it exists.
Šanin’s classical -Lemma (cf. Jech [8, Theorem 9.18]) is the following.
Let be an uncountable collection of finite sets. Then there are an uncountable subset of and a finite set (the root of ) such that for all distinct .
We recall some basic terminology in the theory of forcing, which we shall use systematically in Section 7. A subset in a poset is dense if it meets every principal ideal of . We say that is centred if every finite subset of has a lower bound in . We say that has precaliber if every uncountable subset of has an uncountable centred subset; in particular, this implies the countable chain condition. For a collection of subsets of , a subset of is -generic if for each dense . The following classical lemma (cf. Jech [8, Lemma 14.4]) is, formally, a poset analogue of Baire’s category Theorem.
Let be a countable collection of subsets of a poset . Then each element of is contained in a -generic filter on .
For there may be no -generic filters of (cf. Jech [8, Exercise 16.11]), nevertheless for restricted classes of partial orderings we obtain set-theoretical axioms that are independent of . We shall need the following proper weakening of the instance of Martin’s Axiom usually denoted by either or (cf. Jech [8, Section 16] and Weiss [20, Section 3]); the first parameter refers to the cardinality of .
For every poset of precaliber and every collection of subsets of , if , then there exists a -generic filter on .
3. Simplified morasses
For a positive integer , trying to build certain structures of size as direct limits of countable structures may impose very demanding constraints on the direct systems used for the construction. The pattern of the repetitions of the countable building blocks and their transition morphisms in the direct system is then coded by a complex combinatorial object called a gap- morass. Gap- morasses were introduced by Ronald Jensen in the seventies, enabling him to solve positively the finite gap cardinal transfer conjecture in the constructible universe . The existence of morasses is independent of the usual axiom system of set theory . For example, there are gap- morasses in (cf. Devlin [2, Section VIII.2]), and even in the universe of sets constructible with oracle , for any ; hence if is not inaccessible in , then there is a gap- morass in the ambient set-theoretical universe (cf. Devlin [2, Exercise VIII.6]). Conversely, the existence of a gap- morass implies the existence of a Kurepa tree while, in the generic extension obtained by Levy collapsing an inaccessible cardinal on while preserving , there is no Kurepa tree (cf. Silver ). In particular, the non-existence of a gap- morass is equiconsistent, relatively to , to the existence of an inaccessible cardinal.
However, even for the combinatorial theorems involving morasses are hard to come by, due to the extreme complexity of the definition of gap- morasses. Fortunately, the definition of a gap- morass has been greatly simplified by Dan Velleman , where it is proved that the existence of a gap- morass is equivalent to the existence of a ‘simplified -morass’.
We denote the (noncommutative) ordinal addition by . For ordinals , we denote by the unique ordinal such that . Furthermore, we denote by the order-embedding from ordinals to ordinals defined by
We shall use the following definition, obtained by slightly amending the one in Devlin [2, Section VIII.4] by requiring instead of , but all s nonzero (which could not hold for ). It is easily obtained from a simplified morass as defined in Devlin [2, Section VIII.4] by adding a new zero element to each (that is, by replacing by ) and replacing any by the unique zero-preserving map sending to , for each .
Let be an infinite cardinal. A simplified -morass is a structure
satisfying the following conditions:
, for each , and .
is a set of order-embeddings from into , for all .
, for all .
If , then .
For each , there exists a nonzero ordinal such that and , where denotes the restriction of from into .
For every limit ordinal , all and , for , there exists with together with , for , and such that for each .
The equality holds for each .
It is proved in Velleman  that for regular uncountable, there exists a -morass iff there exists a simplified -morass. For the countable case, the existence of a -morass is provable in , see Velleman .
Simplified morasses as above satisfy the following simple but very useful lemma, which is the basis of the construction of the Kurepa tree obtained from a -morass (cf. Velleman [16, Lemma 3.2]).
Let , let , and let . If , then and .
4. Ladders and breadth
The classical definition of breadth, see Ditor [3, Section 4], runs as follows. Let be a positive integer. A join-semilattice has breadth at most if for every nonempty finite subset of , there exists a nonempty with at most elements such that . This is a particular case of the following definition of breadth, valid for every poset, and, in addition, self-dual: we say that a poset has breadth at most , if for all , () in , if for all in , then there exists such that .
Let be a positive integer. A -ladder is a lower finite lattice in which every element has at most lower covers.
Every -ladder has breadth at most . The diamond has breadth but it is not a -ladder. Every finite chain is a -ladder. The chain of all non-negative integers is also a -ladder. Note that -ladders are called -frames in Dobbertin ; the latter terminology being already used for a completely different lattice-theoretical concept (von Neumann frames), we will not use it. The following is proved in Ditor .
Let be a positive integer. Then every lower finite lattice of breadth at most (thus, in particular, every -ladder) has at most elements.
Proposition 4.2 is especially easy to prove by using Kuratowski’s Free Set Theorem, see Kuratowski . The converse is obviously true for —that is, there exists a -ladder of cardinality (namely, the chain of all natural numbers); also for , by the following result of Ditor , also proved by Dobbertin . We include a proof for convenience.
There exists an atomistic -ladder of cardinality .
We construct inductively a -sequence of countable atomistic -ladders, such that implies that is a proper ideal of . Once this is done, the -ladder will clearly solve our problem.
We take . If is a limit ordinal and all , for , are constructed, take . Suppose that is constructed. If is finite, pick outside objects , and put , with the additional relations and for each . If is infinite, then, as it is a countable lattice, it has a strictly increasing cofinal sequence . Consider a one-to-one sequence of objects outside and put
with the additional relations for and for . The -sequence thus constructed is as required. ∎
The most natural attempt at proving the existence of a -ladder of cardinality , by imitating the proof of Proposition 4.3, would require that every -ladder of cardinality has a cofinal meet-subsemilattice which is also a -ladder (cf. the proof of Theorem 7.9). We do not know whether this statement is a theorem of , although, by Theorem 7.8, it is consistent with . The following example shows that the most straightforward attempt at proving that statement, by expressing structures of cardinality as directed unions of countable structures, fails.
There exists a countable -ladder with an ideal and a cofinal meet-subsemilattice of which is also a -ladder, although there is no cofinal meet-subsemilattice of breadth at most two of containing .
We denote by the lattice represented in Figure 1, and we put
Suppose that there is a cofinal meet-subsemilattice of , with breadth at most two, containing . As is cofinal in , there exists such that . As belongs to , it also belongs to , thus also belongs to , so is contained in , and so, as has breadth at most two, belongs to , a contradiction. ∎
A particular case of the problem above is stated on top of Page 58 in Ditor : let be a -ladder of cardinality . Does the -ladder , endowed with the product order, have a cofinal meet-subsemilattice which is also a -ladder? The answer to that question is affirmative, due to the following easy result.
Let be an infinite, lower finite lattice. Then has a cofinal meet-subsemilattice isomorphic to .
As is an infinite, lower finite lattice, it has a strictly increasing sequence . As is lower finite, for each , there exists a largest natural number such that ; denote this integer by . We define as the graph of , that is, . As is a meet-homomorphism, is a meet-subsemilattice of . Furthermore, defines a lattice isomorphism of onto , and is obviously cofinal in . ∎
5. Large semilattices with bounded breadth
In this section we shall see that weak analogues of -ladders, obtained by replacing by a regular uncountable cardinal and the condition that every element has at most lower covers by the breadth being below , can be easily constructed (cf. Theorem 5.3).
The following Lemma 5.1 is a slightly more general version of [3, Proposition 4.3]. It says that breadth can be verified on the generators of a join-semilattice, the difference with [3, Proposition 4.3] lying in the definition of a generating subset.
A subset in a join-semilattice generates , if every element of is a (not necessarily finite) join of elements of . Equivalently, for all such that , there exists such that and .
Let be a join-semilattice, let be a generating subset of , and let be a positive integer. Then has breadth at most iff for every subset , there exists such that .
We prove the nontrivial direction. Let , and suppose that for each . As generates , there exists such that and , for all . Hence , for each , a contradiction as all elements belong to and by assumption. ∎
An algebraic closure operator on a poset is a map such that (we say that is idempotent), for each , implies that for all , and for each nonempty upward directed subset of admitting a join, the join of exists and is equal to . (We shall refer to the latter property as the join-continuity of .)
For each regular uncountable cardinal and each positive integer , there exists an algebraic closure operator on such that
As is regular uncountable, is a self-map of . It obviously satisfies both (5.1) and (5.2), together with all properties defining an algebraic closure operator except idempotence. Furthermore, for every ,
so . ∎
Therefore, we obtain a positive answer to the specialization of Question A, Page 57 in Ditor  to regular uncountable cardinals.
For each regular uncountable cardinal and each positive integer , there exists an atomistic -semilattice of breadth and cardinality such that for each .
Let be an algebraic closure operator as in Lemma 5.2. We endow with containment. Obviously, is a -semilattice and for each . As , it follows from [3, Theorem 5.2] that has breadth at least . As is nonzero and by (5.2), every singleton , for , belongs to , so is atomistic, and so generates in the sense required by the statement of Lemma 5.1. Therefore, it follows from (5.1) together with Lemma 5.1 that has breadth at most . ∎
For , it it not known whether Theorem 5.3 extends to singular cardinals, for instance (cf. Problem 2 in Ditor ). For , it is not known whether Theorem 5.3 extends to (cf. Problem 1 in Ditor ), although we shall prove in two different ways that a positive answer is consistent with (cf. Theorems 7.9 and 9.1).
6. Preskeletons in normed lattices
A normed lattice is a pair , where is a lattice and is a join-homomorphism from to the ordinals (the ‘norm’). In such a case we put
Observe that each is either empty or an ideal of (we will say that it is an extended ideal of ). We call the subsets the levels of . We say that is transitive if its range, that is, the range of , is an ordinal. Of course, in such a case, .
Observe that conversely, every increasing well-ordered sequence of extended ideals of with union defines a norm , via the rule
In case is lower finite, the set is a finite ideal of , for every and every ordinal , hence it has a largest element, that we shall denote by . The assignment is isotone, that is, and implies that . We shall put
Observe that is a chain. Furthermore, as is a subset of , it is finite. The binary relation defined by the rule
is a partial ordering on in which all principal ideals are finite chains. We shall always denote by the norm function on a normed lattice.
We shall repeatedly use the following easy observation.
The following implications hold, for any elements and in a lower finite normed lattice :
implies that .
(i). From it follows that , hence .
(ii). We need to prove only the direct implication. It is trivial that . For the converse, observe that , thus, as , we obtain that , and so the equality holds. ∎
A preskeleton of a lower finite normed lattice is a subset of satisfying the following conditions:
is a (possibly empty) chain, for every ordinal ;
is projectable, that is, belongs to , for any and any ordinal .
If, in addition, is cofinal in for each , we say that is a skeleton of .
Every preskeleton of a lower finite normed lattice satisfies the following properties:
is a meet-subsemilattice of ;
every element of has at most two lower covers in .
In particular, if is upward directed, then it is a -ladder.
(i). Let and put . Hence . From it follows that , thus . Similarly, . As both and belong to , it follows that they are comparable, and therefore .
(ii). Let . We denote by the largest element of smaller than if it exists (i.e., as is a chain, if has an element smaller than ), and by the largest element of if it exists (i.e., if there exists an element in smaller than ). It suffices to prove that every element smaller than lies either below or below . From it follows that and . If then, as , exists and . If , then, as and by Lemma 6.2(i), , thus exists and .
Now let be upward directed. Together with (i) and the lower finiteness of , this implies that is a lattice. Therefore, by (ii), is a -ladder. ∎
In particular, every skeleton of is both a -ladder and a meet-subsemilattice of . Easy examples show that a skeleton of may not be a join-subsemilattice of .
7. The poset of all finite preskeletons of
As the present section involves Martin’s Axiom in an essential way, we shall use forcing terminology (open, dense) rather than poset terminology (lower subset, coinitial) throughout (cf. Section 2).
It is clear that is an open subset of .
The subset is dense in , for each .
Let , we must find containing . Put. As the finite chain is projectable, the subset is projectable. Furthermore, for each , from it follows that , and so is a preskeleton of .
Put . From it follows that and (cf. Lemma 6.2(i)). Therefore, the element witnesses that belongs to . ∎
A normed lower finite lattice is locally countable if all its levels are countable.
Every locally countable normed lower finite lattice has cardinality at most and range of order-type at most .
Without loss of generality, is an ordinal. For each and each , the map (, ) is finite-to-one, thus, as is countable, is countable as well. As , it follows that . Therefore, has cardinality at most . ∎
In the following lemma, we shall use the -Lemma together with the notions “centred” and “precaliber” (cf. Section 2).
Suppose that is locally countable. Then the poset has precaliber .
Let be an -sequence of elements of , we must find an uncountable such that is centred. Put (an element of ), (a finite subset of ), and (a finite subset of ), for each . Two successive applications of the -Lemma yield an uncountable subset of and finite sets , such that
As is a finite subset of , it is contained in for some . For each , is a finite subset of , hence, as is countable, there are a finite subset of and an uncountable subset of such that
We claim that is centred. It is sufficient to prove that for each nonempty finite subset of , the union belongs to . As is obviously projectable, it suffices to prove that any elements , for , are comparable. Let such that . If then, as is a preskeleton of , we are done. Suppose that . As belongs to (cf. (7.1)), we get , thus, by (7.2), , so , and so, as is a chain, and are comparable. ∎
The following lemma describes -generic filters over .
Let be a lower finite normed lattice and let be a -generic filter of . Then is a skeleton of . In particular, is a cofinal meet-semilattice in and it is a -ladder.
As is an upward directed (for containment) set of preskeletons of , is also a preskeleton of . As meets for each , is cofinal on each level of . In particular, is cofinal in , thus it is upward directed, and thus (cf. Lemma 6.4) it is a -ladder. ∎
Every countable lower finite normed lattice has a skeleton.
As is countable, it has a cofinal chain . It is obvious that the subset is projectable. For any , any two elements of have the form and , for some . As and are comparable, so are and . This proves that is a preskeleton of . As is cofinal in , every lies below some . Hence and, by Lemma 6.2(i), . Therefore, is cofinal in . ∎
For lattices of cardinality we get the following result:
Assume that the axiom holds. Let be a lower finite lattice of cardinality at most . Then has a skeleton.
We obtain the following result.
Suppose that holds. Then there exists an atomistic -ladder of cardinality .
We argue as in the proof of Proposition 4.3. We construct inductively a -sequence of atomistic -ladders of cardinality at most , such that implies that is a proper ideal of . Once this is done, the -ladder will clearly solve our problem.
We take . If is a limit ordinal and all , for , are constructed, take . Let and suppose that is constructed, of cardinality at most . By Theorem 7.8 and Lemma 6.4, has a cofinal meet-subsemilattice which is also a -ladder. We consider an isomorphic copy of disjoint (set-theoretically) from , via an isomorphism , and we consider the partial ordering on obtained as the union of the respective partial orderings of and together with the additional pairs
As is a meet-subsemilattice of , is a lattice. It is easily seen to be an atomistic -ladder (the only atom in not in is ). In particular, the lower covers of , for , are and (in case ) , , where and are the lower covers of in .
The -sequence thus constructed is as required. ∎
8. Amalgamating countable normed -ladders
A poset is the strong amalgam of two subsets and over a subset , if , , and for all , (resp., ) iff there exists such that (resp., ).
The proof of the existence of a -ladder of cardinality from a -morass is based on the following lemma (the maps are defined in (3.1)).
Let be a countable, transitive, normed -ladder with range , let be an ordinal with , and put . Then there are a countable, transitive, normed -ladder with range , containing as an ideal, and a lower embedding such that the following conditions hold:
is the strong amalgam of and over ;
every element of is the join of two elements of ;
is the identity map on ;
The equality holds for each .
It is straightforward to construct a (set-theoretical) copy of such that , isomorphic to via a bijection such that . Defining the ordering on according to Definition 8.1, we obtain that is a strong amalgam of and over . (As , remains the least element of .) We extend the norm to by setting
Observe that the extension of thus defined preserves all finite joins within each block or .
Pick and fix a cofinal chain in with as least element. We set
(cf. (6.1)). Observe that the equality holds for each .
We refer the reader to Section 2 for the notation .
The subset is a cofinal meet-subsemilattice of . Furthermore, the equality holds for each .
Proof of Claim..
As belongs to for each , is cofinal in . Let , we must prove that . We may assume that . It is straightforward to verify that , and so belongs to .
Now let , put and . As and , it follows from Lemma 6.2(i) that , hence belongs to ; it obviously lies above . For each above , it follows from , , and that . From it follows that , thus, as , we get , and thus (the latter equality following from the relation ). Therefore,