Let p be an odd prime. A method is described which given a structure M of finite similarity type produces a nilpotent group of class 2 and exponent p which is in the same stability class as M. Theorem. There are nilpotent groups of class 2 and exponent p in all stability classes. Theorem. The problem of characterizing a stability class is equivalent to characterizing the (nilpotent, class 2, exponent p) groups in that class.

We study descriptive set theory in the space ω1 ω 1 by letting trees with no uncountable branches play a similar role as countable ordinals in traditional descriptive set theory. By using such trees, we get, for example, a covering property for the class of Π 1 1 -sets of ω1 ω 1 . We call a family U of trees universal for a class V of trees if $\mathscr{U} \subseteq \mathscr{V}$ and every tree in V can be order-preservingly mapped (...) into a tree in U. It is well known that the class of countable trees with no infinite branches has a universal family of size ℵ 1 . We shall study the smallest cardinality of a universal family for the class of trees of cardinality ≤ℵ 1 with no uncountable branches. We prove that this cardinality can be 1 (under ¬CH) and any regular cardinal κ which satisfies ℵ 2 ≤ κ ≤ 2 ℵ 1 (under CH). This bears immediately on the covering property of the Π 1 1 -subsets of the space ω1 ω 1 . We also study the possible cardinalities of definable subsets of ω1 ω 1 . We show that the statement that every definable subset of ω1 ω 1 has cardinality $ or cardinality 2 ω1 is equiconsistent with ZFC (if n ≥ 3) and with ZFC plus an inaccessible (if n = 2). Finally, we define an analogue of the notion of a Borel set for the space ω1 ω 1 and prove a Souslin-Kleene type theorem for this notion. (shrink)

It is consistent with ¬CH that every universal theory of relational structures with the joint embedding property and amalgamation for P --diagrams has a universal model of cardinality ℵ 1. For classes with amalgamation for P --diagrams it is consistent that $2^{\aleph_0} > \aleph_2$ and there is a universal model of cardinality ℵ 2.

c.c.c. posets are characterised in terms of N-generic conditions. This characterisation can be applied to get simple proofs of many facts about c.c.c. forcing including $\operatorname{Con}(MA + \neg CH)$.

It is consistent that for many cardinals λ there is a family of at least λ + unbounded subsets of λ which have uniformization properties. In particular if it is consistent that a supercompact cardinal exists, then it is consistent that ℵ ω has such a family. We have applications to point set topology, Whitehead groups and reconstructing separable abelian p-groups from their socles.

We study descriptive set theory in the space $^{\omega_1}\omega_1$ by letting trees with no uncountable branches play a similar role as countable ordinals in traditional descriptive set theory. By using such trees, we get, for example, a covering property for the class of $\Pi^1_1$-sets of $^{\omega_1}\omega_1$. We call a family $\mathscr{U}$ of trees universal for a class $\mathscr{V}$ of trees if $\mathscr{U} \subseteq \mathscr{V}$ and every tree in $\mathscr{V}$ can be order-preservingly mapped into a tree in $\mathscr{U}$. It is well (...) known that the class of countable trees with no infinite branches has a universal family of size $\aleph_1$. We shall study the smallest cardinality of a universal family for the class of trees of cardinality $\leq\aleph_1$ with no uncountable branches. We prove that this cardinality can be 1 (under $\neg$CH) and any regular cardinal $\kappa$ which satisfies $\aleph_2 \leq \kappa \leq 2^{\aleph_1}$ (under CH). This bears immediately on the covering property of the $\Pi^1_1$-subsets of the space $^{\omega_1}\omega_1$. We also study the possible cardinalities of definable subsets of $^{\omega_1}\omega_1$. We show that the statement that every definable subset of $^{\omega_1}\omega_1$ has cardinality $<\omega_n$ or cardinality $2^{\omega_1}$ is equiconsistent with ZFC (if $n \geq 3$) and with ZFC plus an inaccessible (if $n = 2$). Finally, we define an analogue of the notion of a Borel set for the space $^{\omega_1}\omega_1$ and prove a Souslin-Kleene type theorem for this notion. (shrink)

It is consistent with $\neg\mathrm{CH}$ that every universal theory of relational structures with the joint embedding property and amalgamation for $\mathscr{P}^-(3)$-diagrams has a universal model of cardinality $\aleph_1$. For classes with amalgamation for $\mathscr{P}^-(4)$-diagrams it is consistent that $2^{\aleph_0} > \aleph_2$ and there is a universal model of cardinality $\aleph_2$.

A partially ordered set is called k-homogeneous if any isomorphism between k-element subsets extends to an automorphism of . Assuming the set-theoretic assumption ⋄, it is shown that for each k, there exist partially ordered sets of size ϰ1 which embed each countable partial order and are k-homogeneous, but not -homogeneous. This is impossible in the countable case for k ≥ 4.

Most people encounter art images as digital reproductions on a computer screen instead of as originals in a museum or gallery. With the development of digital technologies, high-resolution artworks can be accessed anywhere and anytime by a large number of viewers. Since these digital images depict the same content and are attributed to the same artist as the original, it is often implicitly assumed that their aesthetic evaluation will be similar. When it comes to the digital reproductions of art, however, (...) it is also obvious that reproductions do differ from the originals in various aspects. Besides image quality, resolution, and format, the most obvious change is in the representation of color. The effects of subjectively varying surface-level image features on art evaluation have not been clearly assessed. To address this gap, we compare the evaluation of digital reproductions of 16 expressionist and impressionist paintings manipulated to have a high color saturation vs. a saturation similar to the original. We also investigate the impact of viewing time and expertise, two other aspects that may impact the perception of art in online contexts. Moreover, we link these dimensions to a recent model of aesthetic experience [the Vienna Integrated Model of Top-Down and Bottom-Up Processes in Art Perception ]. Results suggest that color saturation does not exert a major influence on liking. Cognitive and emotional aspects, however, are affected – to different extents for experts and laypersons. For laypersons, the increase in color saturation led to more positive assessments of an artwork, whereas it resulted in increased confusion for art experts. This insight is particularly important when it comes to reproducing artworks digitally. Depending on the intended use, increasing or decreasing the color saturation of the digitally reproduced image might be most appropriate. We conclude with a discussion of these findings and address the question of why empirical aesthetics requires more precise dimensions to better understand the subtle processes that take place in the perception of today’s digitally reproduced art environment. (shrink)

Player motivation is a key research area within games research, with the aim of understanding how the motivation of players is related to their experience and behavior in the game. We present the results of a cross-sectional study with data from 750 players of League of Legends, a popular Multiplayer Online Battle Arena game. Based on the motivational regulations posited by Self-Determination Theory and Latent Profile Analysis, we identify four distinct motivational profiles, which differ with regards to player experience and, (...) to a lesser extent, in-game behavior. While the more self-determined profiles "Intrinsic" and "Autonomous" report mainly positive experience-related outcomes, a considerable part of the player base does not. Players of the "Amotivated" and "External" profile derive less enjoyment, experience more negative affect and tension, and score lower on vitality, indicating game engagement that is potentially detrimental to players' well-being. With regards to game metrics, minor differences in the rate of assists in unranked matches and performance indicators were observed between profiles. This strengthens the notion that differences in experiences are not necessarily reflected in differences in behavioral game metrics. Our findings provide insights into the interplay of player motivation, experience, and in-game behavior, contributing to a more nuanced understanding of player-computer interaction. (shrink)

For every structure M of finite signature Mekler 781) has constructed a group G such that for every κ the maximal number of n -types over an elementary equivalent model of cardinality κ is the same for M and G . These groups are nilpotent of class 2 and of exponent p , where p is a fixed prime greater than 2. We consider stable structures M only and show that M is CM -trivial if and only if G (...) is CM -trivial. Furthermore, we obtain that the free group F 2 in the variety of 2 -nilpotent groups of exponent p>2 with ω free generators has a CM -trivial ω -stable theory. (shrink)

In this paper, we investigate a new model theoretical tree property (TP), called the antichain tree property (ATP). We develop combinatorial techniques for ATP. First, we show that ATP is always witnessed by a formula in a single free variable, and for formulas, not having ATP is closed under disjunction. Second, we show the equivalence of ATP and [Formula: see text]-ATP, and provide a criterion for theories to have not ATP (being NATP). Using these combinatorial observations, we find algebraic examples (...) of ATP and NATP, including pure groups, pure fields and valued fields. More precisely, we prove Mekler’s construction for groups, Chatzidakis’ style criterion for pseudo-algebraically closed (PAC) fields, and the AKE-style principle for valued fields preserving NATP. We give a construction of an antichain tree in the Skolem arithmetic and atomless Boolean algebras. (shrink)

We show that it is consistent with ordinary set theory ZFC and the generalized continuum hypothesis that there exist two 1-separable abelian groups of cardinality 1 which are filtration-equivalent and one is a Whitehead group but the other is not. This solves one of the open problems from Eklof and Mekler [2].

We introduce the large-cardinal notions of ξ-greatly-Mahlo and ξ-reflection cardinals and prove (1) in the constructible universe, L, the first ξ-reflection cardinal, for ξ a successor ordinal, is strictly between the first ξ-greatly-Mahlo and the first Π1ξ-indescribable cardinals, (2) assuming the existence of a ξ-reflection cardinal κ in L, ξ a successor ordinal, there exists a forcing notion in L that preserves cardinals and forces that κ is (ξ+1)-stationary, which implies that the consistency strength of the existence of a (ξ+1)-stationary (...) cardinal is strictly below a Π1ξ-indescribable cardinal. These results generalize to all successor ordinals ξ the original same result of Mekler–Shelah [A. Mekler and S. Shelah, The consistency strength of every stationary set reflects, Israel J. Math.67(3) (1989) 353–365] about a 2-stationary cardinal, i.e. a cardinal that reflects all its stationary sets. (shrink)

Let S and T be countable complete theories. We assume that T is superstable without the dimensional order property, and S is interpretable in T in such a way that every model of S is coded in a model of T. We show that S does not have the dimensional order property, and we discuss the question of whether $\operatorname{Depth}(S) \leq \operatorname{Depth}(T)$ . For Mekler's uniform interpretation of arbitrary theories S of finite similarity type into suitable theories T s (...) of groups we show that $\operatorname{Depth}(S) \leq \operatorname{Depth}(T_S) \leq 1 + \operatorname{Depth}(S)$. (shrink)

We generalize the result of Mekler and Shelah [3] that the existence of a canary tree is independent of ZFC + GCH to uncountable regular cardinals. We also correct an error from the original proof.

We give a solution stated in the title to problem 3 of part 1 of the problems listed in the book of Eklof and Mekler [2], p. 453. There, in pp. 241-242, this is discussed and proved in some cases. The existence of strongly λ-free ones was proved earlier by the criteria in [5] and [3]. We can apply a similar proof to a large class of other varieties in particular to the variety of (non-commutative) groups.

A wide Aronszajn tree is a tree of size and height $\omega _{1}$ with no uncountable branches. We prove that under $MA$ there is no wide Aronszajn tree which is universal under weak embeddings. This solves an open question of Mekler and Väänänen from 1994. We also prove that under $MA$, every wide Aronszajn tree weakly embeds in an Aronszajn tree, which combined with a result of Todorčević from 2007, gives that under $MA$ every wide Aronszajn tree embeds into (...) a Lipschitz tree or a coherent tree. We also prove that under $MA$ there is no wide Aronszajn tree which weakly embeds all Aronszajn trees, improving the result in the first paragraph as well as a result of Todorčević from 2007 who proved that under $MA$ there are no universal Aronszajn trees. (shrink)

We generalize the result of Mekler and Shelah [3] that the existence of a canary tree is independent of ZFC + GCH to uncountable regular cardinals. We also correct an error from the original proof.

We give a solution stated in the title to problem 3 of part 1 of the problems listed in the book of Eklof and Mekler [2], p. 453. There, in pp. 241-242, this is discussed and proved in some cases. The existence of strongly $\lambda$-free ones was proved earlier by the criteria in [5] and [3]. We can apply a similar proof to a large class of other varieties in particular to the variety of (non-commutative) groups.

The paper continues [1]. Let S be a complete theory of ultraflat (e.g. planar) graphs as introduced in [4]. We show a strong form of NOTOP for S: The union of two models M1 and M2, independent over a common elementary submodel M0, is the primary model over M1 ∪ M2 of S. Then by results of [1] Mekler's construction [6] gives for such a theory S of nice ultraflat graphs a superstable 2-step-nilpotent group of exponent $p (>2)$ with (...) NDOP and NOTOP. (shrink)