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Register a new userThe Joint Embedding Property and Maximal Models
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We introduce the notion of a `pure` Abstract Elementary Class to block trivial counterexamples. We study classes of models of bipartite graphs and show: Main Theorem (cf. Theorem 3.5.2 and Corollary 3.5.6): If $(lambda_i : i le alpha<aleph_1)$ is a strictly increasing sequence of characterizable cardinals (Definition 2.1) whose models satisfy JEP$(<lambda_0)$, there is an $L_{omega_1,omega}$ -sentence $psi$ whose models form a pure AEC and (1) The models of $psi$ satisfy JEP$(<lambda_0)$, while JEP fails for all larger cardinals and AP fails in all infinite cardinals. (2) There exist $2^{lambda_i^+}$ non-isomorphic maximal models of $psi$ in $lambda_i^+$, for all $i le alpha$, but no maximal models in any other cardinality; and (3) $psi$ has arbitrarily large models. In particular this shows the Hanf number for JEP and the Hanf number for maximality for pure AEC with Lowenheim number $aleph_0$ are at least $beth_{omega_1}$. We show that although AP$(kappa)$ for each $kappa$ implies the full amalgamation property, JEP$(kappa)$ for each kappa does not imply the full joint embedding property. We show the main combinatorial device of this paper cannot be used to extend the main theorem to a complete sentence.
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The finite models of a universal sentence $Phi$ are the age of a structure if and only if $Phi$ has the joint embedding property. We prove that the computational problem whether a given universal sentence $Phi$ has the joint embedding property is undecidable, even if $Phi$ is additionally Horn and the signature is binary.
In [13] the authors show that if $mu$ is a strongly compact cardinal, $K$ is an Abstract Elementary Class (AEC) with $LS(K)<mu$, and $K$ satisfies joint embedding (amalgamation) cofinally below $mu$, then $K$ satisfies joint embedding (amalgamation) in all cardinals $ge mu$. The question was raised if the strongly compact upper bound was optimal. In this paper we prove the existence of an AEC $K$ that can be axiomatized by an $mathcal{L}_{omega_1,omega}$-sentence in a countable vocabulary, so that if $mu$ is the first measurable cardinal, then (1) $K$ satisfies joint embedding cofinally below $mu$ ; (2) $K$ fails joint embedding cofinally below $mu$; and (3) $K$ satisfies joint embedding above $mu$. Moreover, the example can be generalized to an AEC $K^chi$ axiomatized in $mathcal{L}_{chi^+, omega}$, in a vocabulary of size $chi$, such that (1)-(3) hold with $mu$ being the first measurable above $chi$. This proves that the Hanf number for joint embedding is contained in the interval between the first measurable and the first strongly compact. Since these two cardinals can consistently coincide, the upper bound from [13] is consistently optimal. This is also the first example of a sentence whose joint embedding spectrum is (consistently) neither an initial nor an eventual interval of cardinals. By Theorem 3.26, it is consistent that for any club $C$ on the first measurable $mu$, JEP holds exactly on $lim C$ and everywhere above $mu$.
Soare proved that the maximal sets form an orbit in $mathcal{E}$. We consider here $mathcal{D}$-maximal sets, generalizations of maximal sets introduced by Herrmann and Kummer. Some orbits of $mathcal{D}$-maximal sets are well understood, e.g., hemimaximal sets, but many are not. The goal of this paper is to define new invariants on computably enumerable sets and to use them to give a complete nontrivial classification of the $mathcal{D}$-maximal sets. Although these invariants help us to better understand the $mathcal{D}$-maximal sets, we use them to show that several classes of $mathcal{D}$-maximal sets break into infinitely many orbits.
Based on Hrushovski, Palac{i}n and Pillays example [6], we produce a new structure without the canonical base property, which is interpretable in Baudischs group. Said structure is, in particular, CM-trivial, and thus at the lowest possible level of the ample hierarchy.
An element $a$ of a lattice cups to an element $b > a$ if there is a $c < b$ such that $a cup c = b$. An element of a lattice has the cupping property if it cups to every element above it. We prove that there are non-zero honest elementary degrees that do not have the cupping property, which answers a question of Kristiansen, Schlage-Puchta, and Weiermann. In fact, we show that if $mathbf b$ is a sufficiently large honest elementary degree, then there is a non-zero honest elementary degree $mathbf a <_{mathrm E} mathbf b$ that does not cup to $mathbf b$. For comparison, we modify a result of Cai to show that in sever
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