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Register a new userWhich Classes of Structures Are Both Pseudo-elementary and Definable by an Infinitary Sentence?
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When classes of structures are not first-order definable, we might still try to find a nice description. There are two common ways for doing this. One is to expand the language, leading to notions of pseudo-elementary classes, and the other is to allow infinite conjuncts and disjuncts. In this paper we examine the intersection. Namely, we address the question: Which classes of structures are both pseudo-elementary and $mathcal{L}_{omega_1 omega}$-elementary? We find that these are exactly the classes that can be defined by an infinitary formula that has no infinitary disjunctions.
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For any pair of ordinals $alpha<beta$, $sf CA_alpha$ denotes the class of cylindric algebras of dimension $alpha$, $sf RCA_{alpha}$ denote the class of representable $sf CA_alpha$s and $sf Nr_alpha CA_beta$ ($sf Ra CA_beta)$ denotes the class of $alpha$-neat reducts (relation algebra reducts) of $sf CA_beta$. We show that any class $sf K$ such that $sf RaCA_omega subseteq sf Ksubseteq RaCA_5$, $sf K$ is not elementary, i.e not definable in first order logic. Let $2<n<omega$. It is also shown that any class $sf K$ such that $sf Nr_nCA_omega cap {sf CRCA}_nsubseteq {sf K}subseteq mathbf{S}_csf Nr_nCA_{n+3}$, where $sf CRCA_n$ is the class of completely representable $sf CA_n$s, and $mathbf{S}_c$ denotes the operation of forming complete subalgebras, is proved not to be elementary. Finally, we show that any class $sf K$ such that $mathbf{S}_dsf Ra CA_omega subseteq {sf K}subseteq mathbf{S}_csf RaCA_5$ is not elementary. It remains to be seen whether there exist elementary classes between $sf RaCA_omega$ and $mathbf{S}_dsf RCA_{omega}$. In particular, for $mgeq n+3$, the classes $sf Nr_nCA_m$, $sf CRCA_n$, $mathbf{S}_dsf Nr_nCA_m$, where $mathbf{S}_d$ is the operation of forming dense subalgebras are not first order definable.
We prove a strong non-structure theorem for a class of metric structures with an unstable pair of formulae. As a consequence, we show that weak categoricity (that is, categoricity up to isomorphisms and not isometries) implies severa
Let 0<n^*< omega and f:X-> n^*+1 be a function where X subseteq omega backslash (n^*+1) is infinite. Consider the following set S_f= {x subset aleph_omega : |x| <= aleph_{n^*} & (for all n in X)cf(x cap alpha_n)= aleph_{f(n)}}. The question, first posed by Baumgartner, is whether S_f is stationary in [alpha_omega]^{< aleph_{n^*+1}}. By a standard result, the above question can also be rephrased as certain transfer property. Namely, S_f is stationary iff for any structure A=< aleph_omega, ... > theres a B prec A such that |B|= aleph_{n^*} and for all n in X we have cf(B cap aleph_n)= aleph_{f(n)}. In this paper, we are going to prove a few results concerning the above question.
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Neal J. Evans II
, Mark Heyer. Marc-Antoine Miville-Desch^enes
, Quangn Nguyen-Luong. Manuel Merello
2021
We analyze surveys of molecular cloud structures defined by tracers ranging from CO $J = 1-0$ through $^{13}$CO $J = 1-0$ to dust emission together with NH$_3$ data. The mean value of the virial parameter and the fraction of mass in bound structures depends on the method used to identify structures. Generally, the virial parameter decreases and the fraction of mass in bound structures increases with the effective density of the tracer, the surface density and mass of the structures, and the distance from the center of a galaxy. For the most complete surveys of structures in the Galaxy defined by CO $J = 1-0$, the fraction of mass that is in bound structures is 0.19. For catalogs of other galaxies based on CO $J = 2-1$, the fraction is 0.35. These results offer substantial alleviation of the fundamental problem of slow star formation. If only clouds found to be bound are counted and they are assumed to collapse in a free-fall time at their mean cloud density, the sum over all clouds in a complete survey of the Galaxy yields a predicted star formation rate of 46 solar masses per year, a factor of 6.5 less than if all clouds are bound.
We consider sets of positive integers containing no sum of two elements in the set and also no product of two elements. We show that the upper density of such a set is strictly smaller than 1/2 and that this is best possible. Further, we also find the maximal order for the density of such sets that are also periodic modulo some positive integer.
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