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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Modeloids. I


Author: Miroslav Benda
Journal: Trans. Amer. Math. Soc. 250 (1979), 47-90
MSC: Primary 08A05; Secondary 03C10, 03C68, 03C75, 20B05, 54H20
DOI: https://doi.org/10.1090/S0002-9947-1979-0530044-4
MathSciNet review: 530044
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Abstract: If A is a set and  is the collection of finite nonrepeating sequences of its elements then a modeloid E on A is an equivalence relation on  which preserves length, is hereditary, and is invariant under the action of permutations. The pivotal operation on modeloids is the derivative. The theory of this operation turns out to be very rich with connections leading to diverse branches of mathematics. For example, in §3 we associate an action space with a modeloid and in §5 we characterize the action spaces which are associated with the basic modeloids, i.e., those which are derivatives of themselves. What emerges is a kind of stability for the action space. We then show that action spaces with this stability can be approximated by finite actions and, subject to certain requirements, this approximation is unique (see Proposition 5.7). Algebraically, the countable basic modeloids correspond to closed subgroups of the symmetric groups. This and the study of automorphisms of modeloids let us show, without any algebra, that the only nontrivial normal subgroups of the finite ( $ \left( { \geqslant \,\,5} \right)$) symmetric groups are the alternating groups. The last section gives, hopefully, credence to the thesis that the essence of model theory is the study of modeloids.


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DOI: https://doi.org/10.1090/S0002-9947-1979-0530044-4
Article copyright: © Copyright 1979 American Mathematical Society