Bounded homomorphisms and finitely generated fiber products of lattices

We investigate when fiber products of lattices are finitely generated and obtain a new characterization of bounded lattice homomorphisms onto lattices satisfying a property we call Dean’s condition (D) which arises from Dean’s solution to the word problem for finitely presented lattices. In particular, all finitely presented lattices and those satisfying Whitman’s condition satisfy (D). For lattice epimorphisms [Formula: see text], [Formula: see text], where [Formula: see text], [Formula: see text] are finitely generated and [Formula: see text] satisfies (D), we show the following: If [Formula: see text] and [Formula: see text] are bounded, then their fiber product (pullback) [Formula: see text] is finitely generated. While the converse is not true in general, it does hold when [Formula: see text] and [Formula: see text] are free. As a consequence, we obtain an (exponential time) algorithm to decide boundedness for finitely presented lattices and their finitely generated sublattices satisfying (D). This generalizes an unpublished result of Freese and Nation.

A theorem on partial conservativity in arithmetic

Improving on a result of Arana, we construct an effective family (φr ∣ r ϵ ℚ ∩ [0,1]) of Σn-conservative Πn sentences, increasing in strength as r decreases, with the property that ¬φp is Πn-conservative over PA + φq whenever p < q. We also construct a family of Σn sentences with properties as above except that the roles of Σn and Πn are reversed. The latter result allows to re-obtain an unpublished result of Solovay, the presence of a subset order-isomorphic to the reals in every non-trivial end-segment of every branch of the E-tree, and to generalize it to analogues of the E-tree at higher levels of the arithmetical hierarchy.

On a result of Szemerédi

We provide a short proof that if κ is a regular cardinal with κ < c, thenfor any ordinal α < min{, κ}. In particular,for any ordinal α < . This generalizes an unpublished result of E. Szemerédi that Martin's axiom implies thatfor any cardinal κ with κ < c.

There may be infinitely many near-coherence classes under u < ∂

We show that in the models of u < ∂ from [14] there are infinitely many near-coherence classes of ultrafilters, thus answering Banakh's and Blass' Question 30 of [3] negatively. By an unpublished result of Canjar, there are at least two classes in these models.

Antichains on Three Levels

An antichain is a collection of sets in which no two sets are comparable under set inclusion. An antichain ${\cal A}$ is flat if there exists an integer $k\geq 0$ such that every set in ${\cal A}$ has cardinality either $k$ or $k+1$. The size of ${\cal A}$ is $|{\cal A}|$ and the volume of ${\cal A}$ is $\sum_{A\in{\cal A}}|A|$. The flat antichain theorem states that for any antichain ${\cal A}$ on $[n]=\{1,2,\ldots,n\}$ there exists a flat antichain on $[n]$ with the same size and volume as ${\cal A}$. In this paper we present a key part of the proof of the flat antichain theorem, namely we show that the theorem holds for antichains on three consecutive levels; that is, in which every set has cardinality $k+1$, $k$ or $k-1$ for some integer $k\geq 1$. In fact we prove a stronger result which should be of independent interest. Using the fact that the flat antichain theorem holds for antichains on three consecutive levels, together with an unpublished result by the author and A. Woods showing that the theorem also holds for antichains on four consecutive levels, Á. Kisvölcsey completed the proof of the flat antichain theorem. This proof is to appear in Combinatorica. The squashed (or colex) order on sets is the set ordering with the property that the number of subsets of a collection of sets of size $k$ is minimised when the collection consists of an initial segment of sets of size $k$ in squashed order. Let $p$ be a positive integer, and let ${\cal A}$ consist of $p$ subsets of $[n]$ of size $k+1$ such that, in the squashed order, these subsets are consecutive. Let ${\cal B}$ consist of any $p$ subsets of $[n]$ of size $k-1$. Let $|\triangle_N{\cal A}|$ be the number of subsets of size $k$ of the sets in ${\cal A}$ which are not subsets of any set of size $k+1$ preceding the sets in ${\cal A}$ in the squashed order. Let $|{\bigtriangledown}{\cal B}|$ be the number of supersets of size $k$ of the sets in ${\cal B}$. We show that $|\triangle_N{\cal A}| + |{\bigtriangledown}{\cal B}| > 2 p$. We call this result the 3-levels result. The 3-levels result implies that the flat antichain theorem is true for antichains on at most three, consecutive, levels.

A Simple Approach to Brouwer Degree Based on Differential Forms

We present Heinz’ approach to Brouwer degree in a simpler, shorter and better motivated way. We link it to Kronecker index, use the language of differential forms at an elementary level, and prove the homotopy invariance using an unpublished result of Tartar.

Covering a group with isolators of finitely many subgroups

In [6] B. H. Neumann proved the following beautiful result: if a group G is covered by finitely many cosets, say G = xiHi, then we can omit from the union any xiHi, for which |G|Hj| is infinite. In particular, |G:Hj| is finite, for some j ∈ {l,…,n}.In an unpublished result R. Baer characterized the groups covered by finitely many abelian subgroups, they are exactly the centre-by-finite groups [8]. Coverings by nilpotent subgroups or by Engel subgroups and by normal subgroups have been studied, for example, by R. Baer (see [8]), L. C. Kappe [2,1], M. A. Brodie and R. F. Chamberlain [1], and recently by M. J. Tomkinson [9].

Beth definability in infinitary languages

Some negative results will be proved concerning the following for certain infinitary languages ℒ1 and ℒ2.Definition. Beth(ℒ1, ℒ2) iff, for every sentence ϕ(R) of ℒ1, and n-place relation symbols R and S such that S does not occur in ϕ(R), ifthen there is an ℒ2 formula θ(x1, …, xn) such thatand θ is built up using only those constant and relation symbols of ϕ other than R.That is, Beth(ℒ1, ℒ2) iff for every implicit ℒ1 definition ϕ(R) of relations, there is a corresponding explicit ℒ2 definition θ. Beth(ℒωω, ℒωω) was proved by Beth.Malitz proved that not Beth(ℒω1 ω1, ℒ∞∞) (hence not Beth (ℒ∞∞, ℒ∞∞)), but Beth (ℒ∞ω, ℒ∞∞). In §1, it is shown that Beth(ℒ∞ω, ℒ∞ω) is false. In §2, we strengthen this by showing that, for every cardinal κ, not Beth(ℒ∞ω, ℒ∞κ). In fact, not Beth (ℒκ+ω, ℒ∞κ) follows from property A(κ) defined in §2, and A(κ) is known for regular κ > ω (unpublished result of Morley).More information on infinitary Beth and Craig theorems is given in [2] and [3]. We assume that the reader is acquainted with the languages ℒκλ which allow conjunctions over ≺κ formulas and quantifiers over ≺λ variables. Thus, we assume that the reader is acquainted with the back and forth argument for showing that two structures are ≡∞κ (ℒ∞κ-elementarily equivalent). Our notation is fairly standard.

Second nilpotent BFC groups

The BFC number of a group G is defined to be the least upper bound n of the cardinals of the conjugacy classes of G, provided this is finite, and we then say that G is n-BFC. It was shown by B. H. Neumann [2] that the derived group G′ of such a group is finite, and J. Wiegold [5] proved that.This bound was sharpened by I. D. Macdonald [1] to, and P. M. Neumann has recently communicated the (unpublished) result that G′ ≦ nq(n) with q(n) a quadratic in log2w, an immense improvement on the above. J. A. H. Shepperd and J. Wiegold [4] improved the bound in two special cases, showing that if G is soluble, G′ ≦ np(n) with p(n) a quintic in Iog2n, and that if G is nilpotent of class 2, , It is conjectured that for any n-BFC group G, , Wiegold [5] having shown that this bound is attained by certain nilpotent groups of class 2.