Stability of characters and filters for weighted semilattices

We continue the study of the AMNM property for weighted semilattices that was initiated in Choi (J Aust Math Soc 95(1):36–67, 2013. https://doi.org/10.1017/S1446788713000189). We reformulate this in terms of stability of filters with respect to a given weight function, and then provide a combinatorial condition which is necessary and sufficient for this “filter stability” property to hold. Examples are given to show that this new condition allows for easier and unified proofs of some results in loc. cit., and furthermore allows us to verify the AMNM property in situations not covered by the results of that paper. As a final application, we show that for a large class of semilattices, arising naturally as union-closed set systems, one can always construct weights for which the AMNM property fails.


Background context
This paper is a sequel to [3], in which the first author studied approximately multiplicative functionals on weighted convolution algebras of semilattices, and the problem of whether all such functionals arise as small-norm perturbations of multiplicative ones.
Banach algebras with such a ''stability'' property are said to be AMNM [10] or fstable [9]. When S is a semigroup and x is a (submultiplicative) weight on S, the AMNM condition on the Banach algebra ' 1 x ðSÞ is equivalent to a kind of ''Hyers-Ulam''-type stability for C-valued functions on S, where the stability is measured in terms of the weight x. Hence, one can define the notion of AMNM for the pair ðS; xÞ directly without relying on the Banach-algebraic concept (see Definition 2.6 and Remark 2.7). Indeed, the present paper has been written so that no Banach algebra theory is required.
In the case where S is a semilattice, a sufficient condition for ðS; xÞ to be AMNM was given in [3,Theorem 3.14], which covered many natural examples, such as semilattices with finite breadth. However, it was pointed out by the third author in an unpublished communication, that this condition fails to hold for some straightforward examples, such as Example 3.1 below, and so it was clear that the earlier paper left significant room for improvement. For example, is the weighted semilattice in Example 3.1 AMNM?

New work
In this paper we sharpen some of the techniques introduced in [3], and put them in a more systematic framework, to obtain a necessary and sufficient criterion for a weighted semilattice ðS; xÞ to be AMNM.
As an intermediate step, we observe that it suffices to restrict attention to functions on S with values in f0; 1g, rather than those with codomain C. This was already noted in [3] but we take the opportunity to give some extra details and make some minor corrections. These and other preliminary results are given in Sect. 2, which also has a brief discussion of characters on semilattices and the corresponding filters.
The discretization procedure mentioned above allows us to change perspective from ''character stability'' to ''filter stability'' (with respect to the given weight x). Our criterion is then an intrinsic characterization of the filter stability property, in terms of a combinatorial property of the pair ðS; xÞ that we call propagation. The precise statement is given in Theorem 3.7, and most of Sect. 3 is devoted to setting up the necessary framework.
We highlight two easy applications of Theorem 3.7: (1) It is easy to check that the weighted semilattice described in Example 3.1 has ''propagation at all levels''. Therefore, by Theorem 3.7, this weighted semilattice satisfies filter stability, and hence is AMNM. Details are given in Sect. 3.5, which also explains why the results of [3] are insufficient here. (2) It is also easy to check that if S is a semilattice with finite breadth, then for any choice of weight x we have propagation at all levels. Hence we can apply Theorem 3.7 once again, to deduce that ðS; xÞ is AMNM. This was already observed in [3], but Theorem 3.7 provides a unified framework to see why this result holds.
A third application, which requires substantially more work, forms the final main result of our paper (Theorem 4.8); it addresses a question raised in the closing remarks of [3]. Given the result stated above in (2), and given that there is a weighted semilattice ðT; xÞ which is not AMNM ( [3, Theorem 3.4]), it is natural to ask if every S with infinite breadth can be equipped with some weight x such that ðS; xÞ is not AMNM. While we are unable to resolve this question here, we use the ''propagation'' criterion to obtain a positive answer for a large class of semilattices: namely, those which embed homomorphically into ðP fin ðXÞ; [Þ for some set X (the notation is explained at the start of Sect. 2).

Future work
We would like to extend the construction used to prove Theorem 4.8 to cover all semilattices, not just those S that embed into ðP fin ðXÞ; [Þ. This runs into some serious technical obstacles. Most notably, our construction relies on being able to locate finite free subsemilattices of S that fit togther in a well-behaved way; this is possible here because finite free subsemilattices of ðP fin ðXÞ; [Þ only live on a ''finite part'' of X, and hence can be inductively removed without interfering too much with the embedded copy of S. To move from P fin ðXÞ to the full powerset PðXÞ, it seems that a deeper study is required of union-closed set systems with infinite breadth, and their possible substructures. We intend to pursue this in greater depth in forthcoming work.

Notational conventions
If A and B are non-empty sets then A B denotes the set of functions B ! A; PðBÞ denotes the powerset of B; and P fin ðBÞ denotes the set of all finite subsets of B.
Some of our later examples are subsets of P fin ðXÞ for some non-empty set X. When working with those examples, we shall use upper-case calligraphic letters to denote subsets of P fin ðXÞ, while using letters such as a or E for subsets of X; elements of X will be denoted by lower-case Greek letters.
If S and T are semigroups then the set of homomorphisms from S to T is denoted by HomðS; TÞ. The zero function on a given set will usually be denoted by 0, the domain being clear from context.

Semilattices, characters and filters
We recall some standard definitions, for readers who are not semigroup theorists.
A semilattice is a commutative semigroup S satisfying x 2 ¼ x for all x 2 S. Such an S has a standard and canonical partial order: if x; y 2 S we write x " y whenever xy ¼ x. It is sometimes useful to read x " y as: ''x is a multiple of y'' or ''y is a factor of x''. In this language, xy is the ''largest common multiple'' of x and y.
With respect to ", xy is the meet (or greatest lower bound) of x and y; this gives an alternative, order-theoretic definition of a semilattice, as a poset in which every pair of elements has a meet. Both the semigroup definition and the order-theoretic definition fit well with the intuition from the next example.
Example 2.1 (The 2-element semilattice) The set f0; 1g, equipped with usual multiplication, is a semilattice, which we shall denote by 2. (Note that the partial order defined above satisfies 0 " 1.) If X is a non-empty index set then 2 X , equipped with co-ordinatewise multiplication, is also a semilattice. Each w 2 2 X can be identified with w À1 ð1Þ X, and this defines an isomorphism of semilattices 2 X ffi ðPðXÞ; \Þ.
Semilattices of the form ðPðXÞ; \Þ are universal in a certain sense. Given a semilattice S, for each x 2 S let The definition of " implies that E x \ E y ¼ E xy for all x; y 2 S, and so the function Sch : S ! ðPðSÞ; \Þ defined by x7 !E x is a semigroup homomorphism. Since x 2 E x for all x 2 S, it is also easily verified that Sch is injective. Thus every semilattice can be represented faithfully as a set system which is closed under binary intersectionsor, by taking complements, as a set system which is closed under binary unions. The map Sch has appeared in the literature under various names, such as the Cayley embedding or the Schu¨tzenberger representation [4]. We now turn to characters. Let S be a commutative semigroup, and regard C as a semigroup with respect to multiplication. Following the terminology of [5], a semicharacter on S is a semigroup homomorphism S ! C which is bounded (equivalently, which takes values in the closed unit disc) and which is not identically zero. The set of semicharacters on S is denoted by b S. If S is a semilattice, every semigroup homomorphism S ! C takes values in f0; 1g, so using the notation of Example 2.1 we have b S [ f0g ¼ HomðS; 2Þ 2 S : ð2:2Þ It turns out that b S is a sub-semilattice of 2 S . This is reminiscent of the situation for abelian groups: non-zero semigroup homomorphisms from an abelian group G to C take values in the unit circle T, and it turns out that the set of characters b G is a subgroup of T G . Remark 2.2 (Conflicting terminology) In (semi)lattice theory it is more common to call a non-zero homomorphism S ! 2 a character of the semilattice S. We will adopt this convention, which fits [6,7]; the remarks above show that the concepts of semicharacter and character coincide for semilattices. We shall reserve ''semicharacter'' for general commutative semigroups. Remark 2.3 (Duality theory for semilattices) There is a duality theory for semilattices in which 2 acts as the dualizing object, analogous to Pontryagin duality for (locally compact) abelian groups where T is the dualizing object. See [7] for further details, with an emphasis on category-theoretic aspects; an accessible précis, without proofs, is provided in [6]. Here, we merely remark that there is a natural embedding g : S ! 2 b S which sends x 2 S to b x : /7 !/ðxÞ; this is essentially the same as the Gelfand representation for the commutative Banach algebra ' 1 ðSÞ.
As noted above, f0; 1g-valued functions on S correspond to subsets of S. It is standard knowledge in semilattice theory that, under this correspondence, characters on S (viewed as elements of HomðS; 2Þ) correspond to filters in S. This perspective was already exploited in [3], and will be pursued more systematically in this paper. We therefore review some of the details for the reader's convenience.
Traditionally, a filter in a semilattice S is defined to be a non-empty subset F which is a subsemigroup (so x; y 2 F)xy 2 F) and is ''upwards-closed'' with respect to " (so x 2 S, y 2 F and xy ¼ y )x 2 F). The second condition is equivalent to xy 2 F)x; y 2 F-we leave the proof to the reader-and so for this paper we adopt the following equivalent definition, which makes the link with characters clearer. Definition 2.4 (Filters in semilattices) Let S be a semilattice and let F S. We say that F is a filter in S if it is non-empty and satisfies 8 x; y 2 S ðxy 2 F () x; y 2 FÞ: The set of all filters in S will be denoted by FiltðSÞ.
It follows that for each X S, X is a filter in S if and only if 1 We can use duality theory and the link with filters to give an explanation of the Cayley embedding/Schützenberger representation that was mentioned earlier. Each y 2 S defines a filter F y ¼ ft 2 S : t # yg, and so y7 !1 F y defines an injective function f : S ! b S. By restriction, we obtain a homomorphism of semilattices Checking through the definitions, we find that for each x 2 S f Ã gðxÞ ¼ fy 2 S : b xð1 F y Þ ¼ 1g ¼ fy 2 S : x 2 F y g ¼ fy 2 S : y " xg: Thus f Ã g is exactly the embedding Sch : S ! ðPðSÞ; \Þ from before.

AMNM for weighted semilattices
In this section, we set up the notion of AMNM for weighted semilattices in a way that does not require definitions or results from the theory of Banach algebras. We also take the opportunity to make precise some arguments from [3] that were unclear or incomplete as stated there: see Lemma 2.8 and Proposition 2.9 below. We start in a more general setting. Let S be a semigroup. A weight function on S is a function x : S ! ð0; 1Þ which is submultiplicative; that is, xðxyÞ xðxÞxðyÞ for all x; y 2 S. We shall often refer to the pair ðS; xÞ as a weighted semigroup.
Given a weighted semigroup ðS; xÞ and /; w 2 C S , we define: where in the last definition we view C as a multiplicative semigroup. Note that in all three definitions we allow the value þ1. def x ðwÞ and dist x ðwÞ give two different ways to quantify the failure of w to be a homomorphism; the first has a ''local'' flavour and the second has a ''global'' flavour. We shall sometimes refer to def x ðwÞ as the x -defect of w.
As in [3], we say that w 2 C S is x-bounded if sup x2S jwðxÞjxðxÞ À1 \1. The following lemma is not needed for the eventual application to semilattices, but we include it for sake of completeness and for possible use in future work.
This follows from a technique known in the Banach-algebraic setting [8,Proposition 5.5]. To make this paper more self-contained we provide a full proof, adapted from parts of the proof in [8].
Proof (Jarosz) Let K ¼ sup x2S jwðxÞjxðxÞ À1 , and let s 2 S. Since Taking the supremum over all s yields K 2 À def x ðwÞ K. Since t À 1 t 2 À t for all t ! 0, we obtain K À 1 K 2 À K def x ðwÞ as required. h The lemma may fail if we drop the assumption that w is x-bounded. For instance, take S ¼ N with usual addition, take x 1, and let wðnÞ ¼ 2 n . Then w is clearly not x-bounded, yet def x ðwÞ ¼ 0.
The following terminology is modelled on the corresponding terminology for Banach algebras, as found in [9] or [10].
Definition 2.6 (AMNM/f-stability for weighted semigroups) Let ðS; xÞ be a weighted semigroup. We say that ðS; xÞ has the AMNM property, or is AMNM, or is f-stable, if the following holds: for all e [ 0 there exists d [ 0 such that every xbounded w 2 C S with def x ðwÞ\d satisfies dist x ðwÞ\e.
Remark 2.7 Given a weighted semigroup ðS; xÞ, we can form the associated weighted convolution algebra ' 1 x ðSÞ. Then ðS; xÞ is AMNM if and only if ' 1 x ðSÞ has the Banach-algebraic AMNM property of [10]. The proof of this equivalence is merely an exercise in translating the definitions, together with some standard properties of the ' 1 -norm; details can be found in [3,Sect. 2.2].
We now specialize to the setting where S is a semilattice. Note first that if x is a weight function on a semilattice, we have xðxÞ 2 ! xðx 2 Þ ¼ xðxÞ for all x 2 S, which forces xðxÞ ! 1 for all x 2 S.
Another special feature is that we can improve on Lemma 2.5. The following lemma plugs a small gap in the statement/proof of [3, Corollary 3.7]. In particular, if def x ðwÞ\1 then w is x-bounded.
Proof Without loss of generality, assume def x ðwÞ\1. Let s 2 S. Arguing similarly to the proof of Lemma 2.5, we obtain jwðsÞjxðsÞ À1 ¼ jwðs 2 ÞjxðsÞ À1 ! jwðs 2 ÞjxðsÞ À2 ! jwðsÞ 2 jxðsÞ À2 À def x ðwÞ and hence (again following the proof of the earlier lemma) , d x ðX; YÞ and dist x ð1 X Þ respectively. By the remarks following Definition 2.4, The following result is essentially the same as [3, Corollary 3.7], phrased in different language. However, the proof there was left to the reader, and as mentioned above there was a missing step glossed over. We take this opportunity to provide a slightly weaker but more precise statement, with a complete proof. Proposition 2.9 (Discretizing the AMNM problem for weighted semilattices) Let ðS; xÞ be a weighted semilattice. The following are equivalent: Proof The equivalence of (ii) and (iii) is routine. The implication (i))(ii) follows directly from the definitions and the fact that 1 G is x-bounded for all G S (since x ! 1).
Finally, suppose (ii) holds. Let e [ 0, and let d ¼ dðeÞ be as provided in (ii). Let d 1 [ 0, to be determined later, and let w 2 C S be x-bounded with def x ðwÞ\d 1 . By [3, Lemma 3.6] and the calculations which follow it, there is a set G S such that d x ðw; 1 G Þ\d 1 1=2 with def x ðGÞ 3d 1 1=2 þ 2d 1 . Therefore, provided we originally chose d 1 small enough that d 1 \e 2 and 3d 1 1=2 þ 2d 1 \dðeÞ, condition (ii) implies : Conditions (ii) and (iii) can be viewed as a kind of ''filter stability'' property relative to the weight function x, and for the rest of this paper we shall work exclusively with the perspective of subsets of S rather than functions on S.

Initial remarks
In this section we shall derive an intrinsic, combinatorial characterization of when a given ðS; xÞ has ''stable filters'' in the sense of Proposition 2.9(ii). The precise statement of our characterization is given as Theorem 3.7, and we shall build up to it in stages.
It will be convenient for later examples if we switch to working with log-weights, by which we mean functions k : S ! ½0; 1Þ that satisfy kðxyÞ kðxÞ þ kðyÞ for all x; y 2 S. Given such a k and L ! 0, we define W L ðS; kÞ ¼ fx 2 S : kðxÞ Lg. When there is no danger of confusion we abbreviate this to W L . Example 3.1 Let X be a non-empty set. P fin ðXÞ is a semilattice with respect to binary union. The function x7 !jxj is a log-weight on ðP fin ðXÞ; [Þ, and If we fix r 2 ð1; 1Þ, the function x r : x7 !r jxj is a weight on ðP fin ðXÞ; [Þ.
By varying the ''level'' L we obtain a useful filtration of S. Intuitively, we think of two given sets as ''agreeing at level L'' when they have the same intersection with W L .

Characterizing sets on which dist x is small.
As a first step towards Theorem 3.7, we show that given two subsets E 1 ; E 2 S, d x ðE 1 ; E 2 Þ is small precisely when E 1 and E 2 ''agree at level L'' for some large value of L.
Proof In both parts, the key point is that 1 E 1 ðxÞ À 1 E 2 ðxÞ j jtakes values 0 or 1.
h If E S, let filðEÞ denote the filter-or-empty-set generated by E, i.e. the intersection of all X S such that X 2 FiltðSÞ [ f;g and X E. Note that filðEÞ ¼ ; if and only if E ¼ ;.

Remark 3.3 Let E S and let
Summarizing: if E agrees with some filter-or-empty-set at level L, it agrees with the particular filter-or-empty-set filðE \ W L Þ at level L. Hence, in view of Lemma 3.2, dist x ðEÞ is small if and only if E agrees with filðE \ W L Þ at level L for some large L.

Characterizing sets with small x-defect
Let X S be non-empty. Note that if x; y 2 X and z # xy (that is, z is a factor of xy) then z 2 filðXÞ. Moreover, every z 2 filðXÞ satisfies z # x 1 Á Á Á x k for some x 1 ; . . .; x k 2 X. Definition 3.4 (fbp C -stability) Let C ! 0. For X S we define fbp C ðXÞ :¼ z 2 W C : there exist x; y 2 X \ W C such that z # xy f g : ð3:1Þ (fbp stands for ''factors of binary products''.) Note that fbp C ð;Þ ¼ ;. We always have fbp C ðXÞ X \ W C ; if equality holds, we say that X is fbp c -stable.
X is fbp C -stable if and only if fbp C ðXÞ X: the ''only if'' direction is trivial, and the ''if'' direction holds because fbp C ðXÞ W C for any X S. Lemma 3.5 Let ðS; xÞ be a weighted semilattice, and put k ¼ log x. Let X S.
If def x ðXÞ\d then X is fbp C -stable for any C logðd À1=3 Þ.

Proof
(i) Assume X is fbp C -stable, and let x; y 2 S. We consider two cases. Case A: kðxÞ þ kðyÞ ! C. In this case 1 e ÀC e kðxÞþkðyÞ ¼ e ÀC xðxÞxðyÞ; and so 1 X ðxyÞ À 1 X ðxÞ1 X ðyÞ j j e ÀC xðxÞxðyÞ. Case B: kðxÞ þ kðyÞ C. In this case x, y and xy all belong to W C . Since X is fbp C -stable: • if x and y are in X, then xy 2 fbp C ðfx; ygÞ fbp C ðXÞ X; • if xy lies in X, then both x and y belong to fbp C ðfxygÞ fbp C ðXÞ X.
Thus in this case, 1 X ðxyÞ ¼ 1 X ðxÞ1 X ðyÞ. Putting Case A and Case B together, we see that def x ðXÞ e ÀC .
(ii) If X \ W C ¼ ; then we are done. So suppose X \ W C 6 ¼ ;. Let x; y 2 X \ W C , and let z 2 W C with z # xy. Then and both are strictly less than 1 if def x ðXÞ\d and C logðd À1=3 Þ. Since 1 X ðxÞ ¼ 1 ¼ 1 X ðyÞ, the first formula in (*) implies 1 X ðxyÞ ¼ 1; feeding this into the second formula yields z 2 X. Therefore fbp C ðX \ W C Þ X \ W C , and the converse inclusion is trivial.
h Summarizing: a subset of S has small x-defect precisely when it is fbp Cstable for some large value of C.

fbp c -stability and propagation
An obvious way to obtain fbp C -stable sets is by iteration. Given E S, put fbp 0 C ðEÞ ¼ E \ W C and for k ! 1 recursively define fbp k C ðEÞ ¼ fbp C ðfbp kÀ1 C ðEÞÞ. Then Define fbp 1 C ðEÞ to be the inductive limit S k ! 1 fbp C ðEÞ. By induction, Even when S is finite, the second inclusion in (3.2) can be proper, as we will see in Example 4.2. On the other hand, if E is non-empty and z 2 filðEÞ, there always exists some C ! 0, possibly depending on z, such that z 2 fbp 1 C ðEÞ. (For instance, if This leads naturally to the following definition.
Given L ! 0, we say that ðS; kÞ propagates at level L, or has L-propagation, if It is convenient to set V E ðzÞ :¼ þ1 whenever z 6 2 filðEÞ.
We make some observations for future reference: • For every z 2 S we have V E ðzÞ ! kðzÞ and V E ðzÞ ! inf x2E kðxÞ. (The first inequality holds since fbp 1 C ðEÞ W C . The second one holds because if C\ inf x2E kðxÞ then fbp C ðEÞ ¼ ;, preventing z 2 fbp 1 C ðEÞ.) • If ðS; kÞ propagates at a level L, then it also does so at every lower level. • In the formula defining L-propagation, we could restrict E to the finite subsets of W L without altering the value of the double supremum.
We can now state and prove the promised characterization of ''filter stability''.
Theorem 3.7 Let ðS; xÞ be a weighted semilattice and let k ¼ log x. The following conditions are equivalent.
Proof The equivalence of (i) and (ii) was proved in Proposition 2.9. The equivalence of (ii) and (iii) has been demonstrated in the previous two subsections: specifically, one combines Lemma 3.2, Remark 3.3 and Lemma 3.5 with some standard epsilon-delta book-keeping.
Suppose that condition (iv) holds. Let L ! 0. Then we may choose C ! L such that V G ðzÞ C for all G W L and all z 2 filðGÞ \ W L . Let E S be fbp C -stable and put G ¼ E \ W L . By our choice of C and the definition of V G ðzÞ, But since E is fbp C -stable the right-hand side is contained in E. Hence filðE \ W L Þ \ W L E \ W L , and the converse inclusion is trivial. Thus condition (iii) holds. Conversely, suppose Condition (iii) holds. Let L ! 0 be arbitrary, and choose C ! 0 with the property stated in (iii). Let E W L . Put K ¼ maxðL; CÞ. Since C K, it follows from the definitions that every fbp K -stable set is fbp C -stable. In particular, fbp 1 K ðEÞ is fbp C -stable, so by the property stated in (iii), But since K ! L and E W L , the right-hand side contains filðEÞ \ W L . Hence filðEÞ \ W L fbp 1 K ðEÞ. This shows that the double supremum in (3.4) is K, hence finite; so ðS; kÞ has L-propagation, and (iv) holds. h

An example where our theorem improves on earlier results
In the language of the present paper, [3, Theorem 3.14] may be restated as follows. Let ðS; xÞ be a weighted semilattice, with k ¼ log x, such that for each L ! 0; k is bounded on the subsemilattice generated by W L ðS; kÞ: ðeÞ Then ðS; xÞ is AMNM. Take the semilattice S and log-weight k from Example 3.1. That is, S ¼ ðP fin ðXÞ; [Þ for some nonempty set X, and kðxÞ ¼ jxj for x 2 S. Since fcg 2 W 1 for each c 2 X, the subsemilattice generated by W 1 is fx 2 S : x 6 ¼ ;g. Hence the condition in (Å) fails for L ¼ 1, and [3, Theorem 3.14] cannot be applied. On the other hand, we will now show that ðS; kÞ has L-propagation for all L ! 0. Putting x ¼ exp k, it follows from Theorem 3.7 that ðS; xÞ is AMNM.
Our task will be made slightly simpler using two preliminary observations: • We only need to verify L-propagation for all L 2 N; • We only need to check the supremum in (3.4) for finite subsets of W L .
(For the justification, see the remarks following Definition 3.6.) Let L 2 N, and let E be a non-empty finite subset of W L . Put a ¼ S x2E x, which is finite and non-empty. A straightforward argument shows that filðEÞ ¼ PðaÞ, so that filðEÞ \ W L ¼ z a : jzj L f g : If z a then there exist x 1 ; . . .; x k 2 E such that z x 1 [ . . . [ x k ; moreover, if jzj L then we can always choose k L. Therefore, V E ðzÞ L 2 for all z 2 filðEÞ \ W L . Since this upper bound holds for all finite non-empty E W L , ðS; kÞ propagates at level L, just as we claimed.

Breadth and propagation
The following is a small variation on the previous example.
Example 4.1 (Free semilattices) Let X be a non-empty set, and let P fin Ã ðXÞ denote the set of finite non-empty subsets of X. Equipped with binary union, P fin Ã ðXÞ is a semilattice, which we refer to as the free semilattice generated by X.
This terminology is justified by the following universal property: for any semilattice S and any function f : X ! S, there is a unique homomorphism e f : ðP fin Ã ðXÞ; [Þ ! S that extends f, defined by e f ðaÞ ¼ Q c2a f ðcÞ.
When X 00 is a finite, non-empty set, we write P Ã ðX 00 Þ instead of P fin Ã ðX 00 Þ.  Then k is a log-weight on the semilattice ðP Ã ðX 00 Þ; [Þ. Let E ¼ fxg : x 2 X 00 f g , and note that W 1 ¼ E [ fX 00 g.
Intuitively, for this log-weight, the constraint in the fbp C operation that we can only multiply elements of log-weight C creates a barrier separating us from X 00 , even though X 00 itself has small log-weight.
The previous example suggests that to construct log-weights on a given S for which propagation fails, we should look for isomorphic copies of ðP Ã ðX 00 Þ; [Þ inside S. If E & S is a finite subset, with i E : E ! S being the inclusion map, consider e i E : P Ã ðEÞ ! S, in the notation of Example 4. Remark 4.4 (Comparison with older terminology) Our terminology is not entirely standard; the same property is referred to in [11,12] as ''meet irredundant''. However, for examples which arise as subsemilattices of ðPðXÞ; [Þ, as in the next section, the canonical partial order is not given by inclusion but by containment: a " b () a b; and the ''meet'' of a and b with respect to " is not a \ b, but rather a [ b. In other words, for natural examples we want to consider, the product operation is naturally viewed as a join rather than a meet, and the terminology ''incompressible'' seems more appropriate.

Definition 4.5
The breadth of a semilattice S is defined to be brðSÞ ¼ sup n 2 N : every subset E S with n þ 1 elements is compressible The breadth of a semilattice sheds some light on its structure, and is related to more familiar order-theoretic concepts such as height and width. For instance, suppose brðSÞ ! n. By examining incompressible subsets, one sees that S contains a chain (totally ordered subset) and an antichain (subset in which no two elements are comparable) both of cardinality n; see [1,Sect. 4.1] for further details and some references. In particular, a semilattice S has breadth 1 exactly when the poset ðS; "Þ is totally ordered.
Diverse behaviour occurs even among semilattices of breadth 2. For instance, the following example shows that every infinite k-ary rooted tree (k ! 2) is a semilattice with breadth 2 that contains infinite chains and infinite antichains.
Example 4.6 Let k ! 2. An infinite k-ary rooted tree is an infinite rooted tree in which every vertex has k children. If x and y are vertices in the tree then they have a ''youngest'' common ancestor, which we denote by x^y. Clearly^is a commutative, associative and idempotent binary operation; so the set of vertices becomes a semilattice ðT;^Þ, and the partial order " becomes ''is an ancestor of''.
There is an infinite path P & T obtained by starting at the root and successively taking one of the children; this gives us an infinite chain in ðT; "Þ. If for each element of P we then take one of its children that is not in P, then the collection obtained is an infinite antichain in ðT; "Þ. Let x; y; z 2 T, and let p ¼ x^y^z. Then either x^y or y^z is equal to p: for if not, the set fp; x^y; y^z; yg would form a cycle of length ! 3 in the tree T, which is impossible. Thus every 3-element subset of S is compressible, and so brðSÞ 2. On the other hand, brðSÞ ! 2, since S is not totally ordered.
Remark 4.7 Suppose S is a semilattice with finite breadth. It was originally shown in [3, Example 3.13 and Theorem 3.14] that in this case, ðS; xÞ is AMNM for every weight function x. With the tools of the previous section, we can give an alternative approach to this result. Let n ¼ brðSÞ, and let E be a finite subset of S.
As observed in the remarks before Definition 3.6: if x 1 ; . . .; x k 2 E and z is a factor of x 1 Á Á Á x k , then for any C ! maxf P k i¼1 kðx i Þ; kðzÞg we have z 2 fbp 1 C ðEÞ. But since brðSÞ ¼ n, every z 2 filðEÞ is a factor of x 1 Á Á Á x k for some x 1 ; . . .; x k 2 E with k n. It follows that whenever E W L , V E ðzÞ nL for all z 2 filðEÞ \ W L , and therefore ðS; kÞ propagates at level L for any L ! 0 and any log-weight k. Now we can apply Theorem 3.7.
Motivated by this result, it is natural to ask the following question, which was raised implicitly in [3,Sect. 6].
Question. Let S be a semilattice with infinite breadth. Does there exist a logweight on S for which propagation fails at some level?
At present, we do not have an answer in full generality. However, when S is (isomorphic to) a sub-semilattice of ðP fin ðXÞ; [Þ, the answer is positive. This is the final main result of our paper.  . . .; c n 2 X n a, such that c j 2 b j n b k whenever j 6 ¼ k.
Proof Let M ¼ jaj\1. Since S has infinite breadth it has an incompressible subset of size M þ n, which we may enumerate as b 1 ; . . .; b Mþn . Then for each j, incompressibility implies b j 6 S k:k6 ¼j b k , so T k:k6 ¼j b j n b k is non-empty and we may pick some c j in this set. Note that c 1 ; . . .; c Mþn are all distinct. Since at most M of these belong to a, at least n of them belong to X n a; reordering if necessary, we may take these to be c 1 ; . . .; c n . h Proposition 4.10 There is a sequence ðE n Þ 1 n¼1 in P fin ðXÞ, and a sequence ðF n Þ 1 n¼1 in PðSÞ, with the following properties: • for each n, jE n j ¼ jF n j ¼ n; • the sets E 1 ; E 2 ; . . . are pairwise disjoint; • each x 2 F n satisfies x E j for all 1 j n À 1; • for each n and each c 2 E n , there is a unique x 2 F n such that c 2 x.
Proof We construct both sequences together by (strong) induction on n. For the base case of n ¼ 1: pick any non-empty a 2 S, pick any c 2 a, and let E 1 ¼ fcg, F 1 ¼ fag. Now let n ! 2 and suppose we have found E 1 ; . . .; E nÀ1 and F 1 ; . . .; F nÀ1 with the desired properties. In particular, since each point of E i belongs to some x 2 F i , we have E i joinðF i Þ for i ¼ 1; . . .; n À 1.
Let a :¼ joinðF 1 Þ [ . . . [ joinðF nÀ1 Þ; we have a 2 S since S is closed under finite unions. Now let b 1 ; . . .; b n 2 S and c 1 ; . . .; c n 2 X n a be as provided by Lemma 4.9. For each j, let x j :¼ a [ b j 2 S and note that x j a S nÀ1 i¼1 E i . Also, since c j 6 2 a, whenever j 6 ¼ k we have c j 2 ðb j n b k Þ n a ¼ b j n x k x j n x k : This shows that F n :¼ fx 1 ; . . .; x n g is incompressible (since c j is a point in x j but not in S k:k6 ¼j x k ). Taking E n ¼ fc 1 ; . . .; c n g, the inductive step is complete. h Definition 4.11 (Defining our bad log-weight) Keeping the notation of Proposition 4.10, let D n :¼ S n i¼1 E i for each n 2 N; we also define D 0 :¼ ; and D 1 ¼ S n ! 1 E n . Then, for each x 2 P fin ðXÞ, we make the following definitions: • NðxÞ :¼ max n 2 N 0 : D n x f g ; • rðxÞ :¼ ðx \ D 1 ÞnD NðxÞ ¼ ðxnD NðxÞ Þ \ D 1 ; • gðxÞ :¼ jrðxÞj. h Lemma 4.13 Let n ! 2 and let z n :¼ joinðF n Þ 2 S. Let C ! 0 be such that z n 2 fbp 1 C ðF n Þ. Then 2C ! jE n j ¼ n. Proof Let m 2 N be minimal with the following property: there exists some y 2 fbp m C ðF n Þ such that y E n . This is well-defined, since z n E n (as observed in the proof of Proposition 4.10) and z n 2 fbp 1 C ðF n Þ by assumption. By definition of fbp C , there exist x 1 ; x 2 2 fbp mÀ1 C ðF n Þ such that x 1 [ x 2 y E n . We claim that x i 6 E n for i 2 f1; 2g: • if m ! 2, the claim follows from minimality of m; • if m ¼ 1, the claim holds because fbp 0 C ðF n Þ F n and for each x 2 F n , jx \ E n j ¼ 1 while jE n j ¼ n ! 2.
Proof For n ! 2, each x 2 F n satisfies x \ D 1 ¼ D nÀ1 [ fcg for some c 2 E n . Therefore F n W 1 . Now let z n ¼ joinðF n Þ 2 filðF n Þ. By the properties listed in Proposition 4.10, z n \ D 1 ¼ D n , and gðz n Þ ¼ 0.