408:, edited by D. Haskell et al., Math. Sci. Res. Inst. Publ. 39, Cambridge Univ. Press, New York, 2000. Contains a formal definition of Morley rank.
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265:. A formula defining a finite set has Morley rank 0. A formula with Morley rank 1 and Morley degree 1 is called
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or −1 or ∞, defined by first recursively defining what it means for a formula to have Morley rank at least
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419:, edited by D. Haskell et al., Math. Sci. Res. Inst. Publ. 39, Cambridge Univ. Press, New York, 2000.
226:-saturated models the Morley rank of a subset is the Morley rank of any formula defining the subset.
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The empty set has Morley rank −1, and conversely anything of Morley rank −1 is empty.
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is strongly minimal. Morley rank and strongly minimal structures are key tools in the proof of
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A subset has Morley rank 0 if and only if it is finite and non-empty.
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397:, "Groups of finite Morley rank", Oxford Univ. Press (1994)
183: + 1, and is defined to be ∞ if it is at least
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has countably infinitely many disjoint definable subsets
206:) the Morley rank is defined to be the Morley rank of
339:of maximal dimension; this is not the same as its
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354:, has Morley rank ∞, as it contains a countable
423:Morley, M.D. (1965), "Categoricity in power",
273:structure is one where the trivial formula
358:of definable subsets isomorphic to itself.
285:and in the larger area of model theoretic
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171:The Morley rank is then defined to be
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413:Model Theory of Differential Fields
191:, and is defined to be −1 if
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417:Model theory, algebra and geometry
406:Model theory, algebra and geometry
249: < ω subsets of rank
198:For a definable subset of a model
70:definable (with parameters) subset
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402:Stability theory and its variants
95:The Morley rank is at least 0 if
64:. The Morley rank of a formula
151:, the Morley rank is at least
110:, the Morley rank is at least
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464:"Group of finite Morley rank"
435:American Mathematical Society
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283:Morley's categoricity theorem
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341:degree in algebraic geometry
245:breaks up into no more than
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487:Encyclopedia of Mathematics
469:Encyclopedia of Mathematics
404:(2000) pp. 131–148 in
374:Group of finite Morley rank
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323:, then the Morley rank of
318:algebraically closed field
415:(2000) pp. 53–63 in
369:Cherlin–Zilber conjecture
327:is the same as its usual
218:elementary extension of
136:, each of rank at least
480:Pillay, Anand (2001) ,
426:Trans. Amer. Math. Soc.
331:. The Morley degree of
337:irreducible components
202:(defined by a formula
222:. In particular for ℵ
140: − 1.
26:Michael D. Morley
116:elementary extension
350:, considered as an
175:if it is at least
155:if it is at least
46:algebraic geometry
18:mathematical logic
391:Alexandre Borovik
335:is the number of
187:for all ordinals
179:but not at least
108:successor ordinal
87:for some ordinal
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348:rational numbers
287:stability theory
271:strongly minimal
267:strongly minimal
257:is said to have
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437:: 514–538,
352:ordered set
147:a non-zero
114:if in some
68:defining a
22:Morley rank
385:References
195:is empty.
163:less than
125:, the set
52:Definition
492:EMS Press
474:EMS Press
462:(2001) ,
395:Ali Nesin
316:, for an
237:has rank
233:defining
216:saturated
502:Category
400:B. Hart
363:See also
293:Examples
210:in any ℵ
159:for all
453:1994188
253:, then
81:ordinal
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379:U-rank
308:is an
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433:(2),
269:. A
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346:The
143:For
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312:in
304:If
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