On countable stable structures which are homogeneous for a finite relational language |
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Authors: | A H Lachlan |
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Institution: | (1) Institute for Advanced Studies, The Hebrew University of Jerusalem, Jerusalem, Israel;(2) Present address: Department of Mathematics, Simon Fraser University, V5A 1S6 Burnaby, B. C., Canada |
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Abstract: | LetL be a finite relational language andH(L) denote the class of all countable structures which are stable and homogeneous forL in the sense of Fraissé. By convention countable includes finite and any finite structure is stable. A rank functionr :H(L) →ω is introduced and also a notion of dimension for structures inH(L). A canonical way of shrinking structures is defined which reduces their dimensions. The main result of the paper is that
anyM ∈H(L) can be shrunk toM′ ∈H(L),M′ ⊆M, such that |M′| is bounded in terms ofr(M), and the isomorphism type ofM overM′ is uniquely determined by the dimensions ofM. Forr<ω we deduce thatH(L, r), the class of allM ∈H(L) withr(M)≦r, is the union of a finite number of classes within each of which the isomorphism type of a structure is completely determined
by its dimensions.
Dedicated to the memory of Abraham Robinson on the tenth anniversary of his death |
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