Antichains Of Size Research Articles

On the structure of Borel ideals in-between the ideals [formula omitted] and Fin ⊗ Fin in the Katětov order

For a family F⊆ωω we define the ideal I(F) on ω×ω to be the ideal generated by the family{A⊆ω×ω:∃f∈F∀∞n(|{k:(n,k)∈A}|≤f(n))}. Using ideals of the form I(F), we show that the structure of Borel ideals in-between two well known Borel idealsED={A⊆ω×ω:∃m∀∞n(|{k:(n,k)∈A}|<m))} andFin⊗Fin={A⊆ω×ω:∀∞n(|{k:(n,k)∈A}|<ℵ0))} in the Katětov order is fairly complicated. Namely, there is a copy of P(ω)/Fin in-between ED and Fin⊗Fin, and consequently there are increasing and decreasing chains of length b and antichains of size c.

Erdős–Ulam ideals vs. simple density ideals

The main aim of this paper is to bridge two directions of research generalizing asymptotic density zero sets. This enables to transfer results concerning one direction to the other one. Consider a function g:ω→[0,∞) such that limn→∞⁡g(n)=∞ and ng(n) does not converge to 0. Then the family Zg={A⊆ω:limn→∞⁡card(A∩n)g(n)=0} is an ideal called simple density ideal (or ideal associated to upper density of weight g). We compare this class of ideals with Erdős–Ulam ideals. In particular, we show that there are ⊑-antichains of size c among Erdős–Ulam ideals which are and are not simple density ideals (in [12] it is shown that there is also such an antichain among simple density ideals which are not Erdős–Ulam ideals). We characterize simple density ideals which are Erdős–Ulam as those containing the classical ideal of sets of asymptotic density zero. We also characterize Erdős–Ulam ideals which are simple density ideals. In the latter case we need to introduce two new notions. One of them, called increasing-invariance of an ideal I, asserts that given B∈I and C⊆ω with card(C∩n)≤card(B∩n) for all n, we have C∈I. This notion is inspired by [3] and is later applied in [12] for a partial solution of [15, Problem 5.8]. Finally, we pose some open problems.

THE TUKEY ORDER ON COMPACT SUBSETS OF SEPARABLE METRIC SPACES

AbstractOne partially ordered set, Q, is a Tukey quotient of another, P, if there is a map (a Tukey quotient) $\phi :P \to Q$ carrying cofinal sets of P to cofinal sets of Q. Two partial orders which are mutual Tukey quotients of each other are said to be Tukey equivalent. Let ${\cal D}_{\rm{&amp;#xF01C;}} $ be the partially ordered set of Tukey equivalence classes of directed sets of size $ \le {\rm{&amp;#xF01C;}}$. It is shown that ${\cal D}_{\rm{&amp;#xF01C;}} $ contains an antichain of size $2^{\rm{&amp;#xF01C;}} $, and so has size $2^{\rm{&amp;#xF01C;}} $. The elements of the antichain are of the form ${\cal K}\left( M \right)$, the set of compact subsets of a separable metrizable space M, ordered by inclusion. The order structure of such ${\cal K}\left( M \right)$’s under Tukey quotients is investigated. Relative Tukey quotients are introduced. Applications are given to function spaces and to the complexity of weakly countably determined Banach spaces and Gul’ko compacta.

Large Sets of Lattices without Order Embeddings

Let I and μ be an infinite index set and a cardinal, respectively, such that |I| ≤ μ and, starting from ℵ0, μ can be constructed in countably many steps by passing from a cardinal λ to 2λ at successor ordinals and forming suprema at limit ordinals. We prove that there exists a system X = {Li: i ∈ I} of complemented lattices of cardinalities less than |I| such that if i, j ∈ I and φ: Li → Lj is an order embedding, then i = j and φ is the identity map of Li. If |I| is countable, then, in addition, X consists of finite lattices of length 10. Stating the main result in other words, we prove that the category of (complemented) lattices with order embeddings has a discrete full subcategory with |I| many objects. Still in other words, the class of these lattices has large antichains (that is, antichains of size |I|) with respect to the quasiorder “embeddability.” As corollaries, we trivially obtain analogous statements for partially ordered sets and semilattices.

Degree bounds for linear discrepancy of interval orders and disconnected posets

Let P be a poset in which each point is incomparable to at most Δ others. Tanenbaum, Trenk, and Fishburn asked in a 2001 paper if the linear discrepancy of such a poset is bounded above by ⌊ ( 3 Δ − 1 ) / 2 ⌋ . This paper answers their question in the affirmative for two classes of posets. The first class is the interval orders, which are shown to have linear discrepancy at most Δ , with equality precisely for interval orders containing an antichain of size Δ + 1 . The stronger bound is tight even for interval orders of width 2. The second class of posets considered is the disconnected posets, which have linear discrepancy at most ⌊ ( 3 Δ − 1 ) / 2 ⌋ . This paper also contains lemmas on the role of critical pairs in linear discrepancy as well as a theorem establishing that every poset contains a point whose removal decreases the linear discrepancy by at most 1.

On the structure of the Medvedev lattice

AbstractWe investigate the structure of the Medvedev lattice as a partial order. We prove that every interval in the lattice is either finite, in which case it is isomorphic to a finite Boolean algebra, or contains an antichain of size . the size of the lattice itself. We also prove that it is consistent with ZFC that the lattice has chains of size . and in fact that these big chains occur in every infinite interval. We also study embeddings of lattices and algebras. We show that large Boolean algebras can be embedded into the Medvedev lattice as upper semilattices, but that a Boolean algebra can be embedded as a lattice only if it is countable. Finally we discuss which of these results hold for the closely related Muchnik lattice.

Antichains in partially ordered sets of singular cofinality

In their paper from 1981, Milner and Sauer conjectured that for any poset $$\langle P,\le\rangle$$ , if $$cf(P,\le)=\lambda>cf(\lambda)=\kappa$$ , then P must contain an antichain of size κ. We prove that for λ > cf(λ) = κ, if there exists a cardinal μ < λ such that cov(λ, μ, κ, 2) = λ, then any poset of cofinality λ contains λ κ antichains of size κ. The hypothesis of our theorem is very weak and is a consequence of many well-known axioms such as GCH, SSH and PFA. The consistency of the negation of this hypothesis is unknown.

Degrees of monotone complexity

AbstractLevin and Schnorr (independently) introduced the monotone complexity, Km (α), of a binary string α. We use monotone complexity to define the relative complexity (or relative randomness) of reals. We define a partial ordering ≤Km on 2ω by α ≤Km β iff there is a constant c such that Km(α | n) ≤ Km(β | n)+ c for all n. The monotone degree of α is the set of all β such that α Km β and β Km α. We show the monotone degrees contain an antichain of size , a countable dense linear ordering (of degrees of cardinality ), and a minimal pair.Downey, Hirschfeldt, LaForte, Nies and others have studied a similar structure, the K-degrees, where K is the prefix-free Kolmogorov complexity. A minimal pair of K-degrees was constructed by Csima and Montalban. Of particular interest are the noncomputable trivial reals, first constructed by Solovay. We defineareal to be (Km,K)-trivial if for some constant c, Km(α | n) ≤ K(n) + c for all n. It is not known whether there is a Km-minimal real, but we show that any such real must be (Km,K)-trivial.Finally, we consider the monotone degrees of the computably enumerable (c.e.) and strongly computably enumerable (s.c.e.) reals. We show there is no minimal c.e. monotone degree and that Solovay reducibility does not imply monotone reducibility on the c.e. reals. We also show the s.c.e. monotone degrees contain an infinite antichain and a countable dense linear ordering.

Ordering MAD families a la Katětov

AbstractAn ordering (≤K) on maximal almost disjoint (MAD) families closely related to destructibility of MAD families by forcing is introduced and studied. It is shown that the order has antichains of size c and decreasing chains of length c+ bellow every element. Assuming t = c a MAD family equivalent to all of its restrictions is constructed. It is also shown here that the Continuum Hypothesis implies that for every ωω-bounding forcing ℙ of size c there is a Cohen-destructible, ℙ-indestructible MAD family. Finally, two other orderings on MAD families are suggested and an old construction of Mrówka is revisited.

There are no infinite order polynomially complete lattices, after all

If L is a lattice with the interpolation property whose cardinality is a strong limit cardinal of uncountable cofinality, then some finite power $L^n$ has an antichain of size $\kappa$ . Hence there are no infinite opc lattices. However, the existence of strongly amorphous sets implies (in ZF) the existence of infinite opc lattices.

Chains and antichains in interval algebras

AbstractLet κ be a regular cardinal, and let B be a subalgebra of an interval algebra of size κ. The existence of a chain or an antichain of size κ in ℬ is due to M. Rubin (see [7]). We show that if the density of B is countable, then the same conclusion holds without this assumption on κ. Next we also show that this is the best possible result by showing that it is consistent with 2ℵ0 = ℵω1 that there is a boolean algebra B of size ℵω1 such that length(B) = ℵω1 is not attained and the incomparability of B is less than ℵω1. Notice that B is a subalgebra of an interval algebra. For more on chains and antichains in boolean algebras see. e.g., [1] and [2].

A length-width inequality for partially ordered sets with two-element cutsets

We show that if P is a partially ordered set of width n, and A is an antichain of size n whose elements all have cutsets of size at most 2, then every maximal chain of P has at least n − 2 elements. We also give an extension to larger cutsets.

Changing the Depth of an Ordered Set by Decomposition

The depth of a partially ordered set $\langle P, < \rangle$ is the smallest ordinal $\gamma$ such that $\langle P, < \rangle$ does not embed ${\gamma ^\ast }$. The width of $\langle P, < \rangle$ is the smallest cardinal number $\mu$ such that there is no antichain of size $\mu + 1$ in $P$. We show that if $\gamma > \omega$ and $\gamma$ is not an infinite successor cardinal, then any partially ordered set of depth $\gamma$ can be decomposed into $\operatorname {cf}(|\gamma |)$ parts so that the depth of each part is strictly less than $\gamma$. If $\gamma = \omega$ or if $\gamma$ is an infinite successor cardinal, then for any infinite cardinal $\lambda$ there is a linearly ordered set of depth $\gamma$ such that for any $\lambda$-decomposition one of the parts has the same depth $\gamma$. These results are used to solve an analogous problem about width. It is well known that, for any cardinal $\lambda$, there is a partial order of width $\omega$ which cannot be split into $\lambda$ parts of finite width. We prove that, for any cardinal $\lambda$ and any infinite cardinal $\nu$, there is a partial order of width ${\nu ^ + }$ which cannot be split into $\lambda$ parts of smaller width.