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Projective Limits and Ultraproducts of Nonabelian Finite Groups [article]

Nazih Nahlus, Yilong Yang
2021 arXiv   pre-print
This has led to the investigations of the subgroups of metric ultraproducts of finite groups. This paper attempts to study the dual problem: what are the quotients of ultraproducts of finite groups?  ...  Since an ultraproduct is an abstract quotient of the direct product, this also led to a more general question: what are the abstract quotients of profinite groups?  ...  The ultraproduct functor preserves all finite limits.  ... 
arXiv:2107.09900v1 fatcat:jak7m6o6z5abve725hyci35t3a

Ultraproducts, p-limits and antichains on the Comfort group order

A.H. Tomita, S. Watson
2004 Topology and its Applications  
Ultraproducts have been defined by Lǒs in 1955.  ...  By hypothesis, there exists, for each ξ ∈ F , z ξ ∈ Φ([c] <ω ) such that z ξ is the p-limit of the sequence {Φ(g ξ (n)): n ∈ ω}. We claim that ξ ∈F z ξ is the p-limit of the sequence {y n : n ∈ ω}.  ... 
doi:10.1016/j.topol.2004.02.012 fatcat:d7uykjasl5h7rn5riq4l763flq

Ultraproducts and higher order models

Wilfred Gordon Malcolm
1973 Bulletin of the Australian Mathematical Society  
in which these find expression in the ultraproduct construction.  ...  Model theoretic results paralleling those of a first order theory and ultraproducts are established.  ...  Seelye. 315 316 Wi If red Gordon Malcolm an ultraproduct of a local family of its substructures are proved. Brief mention is given to the presence of 'inverse limits' in this embedding context.  ... 
doi:10.1017/s0004972700042568 fatcat:fi4p3rmmrrhynhvjyu6nh2ypcu

Page 649 of Mathematical Reviews Vol. , Issue 82b [page]

1982 Mathematical Reviews  
It thus provides the theory of nonstandard hulls with immediate contact with constructions in, for example, probability theory that are far from obvious when one’s view is limited solely to the ultraproduct  ...  Consider an ultraproduct Y of a family [ £;:i€J] of Banach spaces and let Z be the corresponding ultraproduct of their dual spaces [E* :i€/].  ... 

Ultraproducts and metastability [article]

Jeremy Avigad, José Iovino
2013 arXiv   pre-print
is the D-limit of its truth values in the individual structures.  ...  The constructions above, however, are not ultraproducts in the first-order sense, since we restrict to "finite" elements, mod out by infinitesimal proximity ∼, and (implicitly, by taking limits with respect  ... 
arXiv:1301.3063v4 fatcat:pguw75hjczdwhaapzs33dc4dzi

Metric ultraproducts of finite simple groups [article]

Andreas Thom, John S. Wilson
2014 arXiv   pre-print
Then G is isomorphic to an ultraproduct of alternating groups or to an ultraproduct of finite simple classical groups.  ...  The isomorphism type of G determines which of these two cases arises, and, in the latter case, the ω-limit of the characteristics of the groups Gi.  ...  and so the metric ultraproduct is a simple group [9] .  ... 
arXiv:1402.0341v1 fatcat:6uocg7pzcfhvtp3bl5khsi4iby

Remarks on compactifications of pseudofinite groups

Anand Pillay
2017 Fundamenta Mathematicae  
Basically referring to results in the literature we point out that (i) the Bohr compactification of an ultraproduct of finite simple groups is trivial, and (ii) the "definable" Bohr compactification of  ...  If bG is the inverse limit of a directed system (L i ) i of compact Lie groups, then clearly (bG) 0 is the inverse limit of the L 0 i .  ...  Now def M bG is an inverse limit of compact Lie groups L i and we want to show that L 0 i is commutative for each i.  ... 
doi:10.4064/fm275-7-2016 fatcat:akkguoqijrgovdt7lfkzlmjl3i

Remarks on compactifications of pseudofinite groups [article]

Anand Pillay
2015 arXiv   pre-print
Basically referring to results in the literature we point out (i) the Bohr compactification of an ultraproduct of finite simple groups is trivial, and (ii) the "definable" Bohr compactification of any  ...  If bG is the inverse limit of a directed system (L i ) i of compact Lie groups, then clearly (bG) 0 is the inverse limit of the L 0 i .  ...  Now def M bG is an inverse limit of compact Lie groups L i and we want to show that L 0 i is commutative for each i.  ... 
arXiv:1509.02895v1 fatcat:abtqusnb7bhnnfrygi5s3qqgzq

Some remarks on finitarily approximable groups [article]

Nikolay Nikolov, Jakob Schneider, Andreas Thom
2017 arXiv   pre-print
Moreover, we discuss the question which connected Lie groups can be embedded into a metric ultraproduct of finite groups with invariant length function.  ...  Now it is clear that G, as the inverse limit of the groups π J (G) (J ⊆ I finite) and the maps π J (G) → π J ′ (G), contains the inverse limit H of the socles of these groups together with the restricted  ...  Metric ultraproducts of groups.  ... 
arXiv:1703.06092v2 fatcat:kflhptymyngcdhevbjl7jeq4cy

Metamathematics of modal logic

Robert Ian Goldblatt
1974 Bulletin of the Australian Mathematical Society  
The general concern of this thesis is to explore the relationships between these two semantical frameworks and to discuss their relative strengths and limitations.  ...  Inverse limits of descriptive frames are defined in Section 11, and used in Section 12 in a characterisation, in terms of closure under various constructions, of those classes of descriptive frames that  ... 
doi:10.1017/s0004972700041186 fatcat:7f6wejp3y5cn7l3gns75ysz5pe

The problem of completeness for Gromov–Hausdorff metrics on C*-algebras

Daniele Guido, Tommaso Isola
2006 Journal of Functional Analysis  
Conditions ensuring the existence of a C*-structure on the limit are considered, making use of the notion of ultraproduct.  ...  This is shown by exhibiting a Cauchy sequence whose limit, which always exists as an operator system, is not completely order isomorphic to any C*-algebra.  ...  Furthermore we show that Cauchy sequences are uniformly compact and the Lip-ultraproduct is indeed the limit.  ... 
doi:10.1016/j.jfa.2005.04.007 fatcat:den3li56o5c4lm5nek2zqjradq

Page 1026 of Mathematical Reviews Vol. 34, Issue 5 [page]

1967 Mathematical Reviews  
The author’s aim is to give an overall view of the work done in ultraproducts and of the part which ultraproducts play in the foundations of mathematics.  ...  of M is not an element of *M,,. | It is, of course, not surprising that some properties of V, | are most easily expressed in terms of the bigger structure | *M_.,; cf. the use of (possibly irrational) limits  ... 

Ultraproducts, weak equivalence and sofic entropy [article]

Alessandro Carderi
2015 arXiv   pre-print
Using this topology, we will show that a profinite action is weakly equivalent to an ultraproduct of finite actions.  ...  We will give a simpler proof of the compactness of the space, showing that convergence is characterized by ultraproducts.  ...  We now show that the ultraproduct of a sequence of actions defined in Definition 2.20 is the limit with respect to the ultrafilter u for the WC-topology.  ... 
arXiv:1509.03189v1 fatcat:sqm53pmuynfsdlyhpjjwrsalla

On the norm of products of polynomials on ultraproduct of Banach spaces [article]

Jorge Tomás Rodríguez
2014 arXiv   pre-print
The purpose of this article is to study the problem of finding sharp lower bounds for the norm of the product of polynomials in the ultraproducts of Banach spaces (X_i)_ U.  ...  Later on, we are going to need the next basic Lemma about limits of ultraproducts, whose proof is an easy exercise of basic topology and ultrafilters. Lemma 2.3.  ...  Then, the limit of (x i ) i∈I respect of U exists and is unique.  ... 
arXiv:1411.5894v1 fatcat:xjgcdpgh3zeafo5zffinjpxg54

On the Commutativity of Ultraproducts with Direct Products

Michel Hebert
1991 Proceedings of the American Mathematical Society  
We show that an ultraproduct of direct products of structures is elementarily equivalent to a direct product (naturally defined over an ultraproduct of sets!) of ultraproducts of these structures.  ...  Let us consider an ultraproduct nD(ily(/)), where x'(i) £Fl¡j'i .  ...  This can be checked directly (the isomorphism being the obviously defined one), or seen as a consequence of a very general algebraic property; namely, that "filtered colimits commute with finite limits  ... 
doi:10.2307/2048461 fatcat:zrva2izxw5hp3fm23jzdjixpo4
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