On the fine structure of the polygroup blow-up

Ben-Yaacov, Itay

doi:10.1007/s00153-003-0173-3

Archive for Mathematical Logic

2003

DOI: 10.1007/s00153-003-0173-3

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On the fine structure of the polygroup blow-up

Itay Ben-Yaacov¹

Abstract: Abstract. We study in detail the blow-up procedure described in [BTW01]. We obtain a structure theorem for coreless polygroups as a double quotient space G/ /H, and a polygroup chunk theorem. Seeking to remove the arbitrary parameter needed for the blow-up, we find canonical ∅-invariant groupoids G > H analogous to G and H above, and show that H contains precisely all the arbitrary choices related to the blow-up.

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Cited by 5 publications

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“…(ii) If R is a β I,f,1 -equivalence relation then every R-class can be covered by ν sets of the form a R1 . Conversely, if R is a γ I,f -equivalence relation having this property, then it is β I,f,f(1) .…”

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Simple Almost Hyperdefinable Groups

Ben-Yaacov

2006

J. Math. Log.

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i) We lay down the groundwork for the treatment of almost hyperdefinable groups: notions from [5] are put into a natural hierarchy, and new notions, essential to the study to such groups, fit elegantly into this hierarchy. (ii) We show that "classical" properties of definable and hyperdefinable groups in simple theories can be generalised to this context. In particular, we prove the existence of stabilisers of Lascar strong types and of the connected and locally connected components of subgroups, and that in a simple one-based theory an almost hyperdefinable group is bounded-by-abelian-by-bounded. . Downloaded from www.worldscientific.com by UNIVERSITY AT BUFFALO on 02/03/15. For personal use only.70 I. Ben-Yaacov find ourselves working in a rather weird category, where "obvious" notions such as intersection can be somewhat surprising.The other aspect is actually proving properties of α/β-groups and their subgroups. While doing so, we shall try to skip tedious step-by-step verifications in this new context of proofs already given in [7,8], if the adaptation of the existing proofs is sufficiently trivial. We rather wish to concentrate on new ideas, such as methods to recover α-elements and α/β-groups from such β/β-groups, using stratified local ranks and the intermediary notion of an α − /β-group.Finally, a word (or several) about our terminology, compared with that of [5] and before. Originally, one considered definable groups, or at most type-definable groups, which lived in the real or in an imaginary sort. Then in [7], one starts considering groups living in hyperimaginary sorts, namely in quotients by typedefinable equivalence relations. In a hyperimaginary sort the distinction between a type-definable and a definable set is meaningless, so we call them hyperdefinable. The next level comes in [5], where we have to replace the type-definable equivalence relation by which we divide by an almost type-definable one, namely a "nice" union of type-definable relations. The quotient is named almost hyperdefinable, in analogy with hyperdefinable. So in "almost type-definable", the "almost" qualifies the numerator (in fact there is not necessarily a denominator), whereas in "almost hyperdefinable" it qualifies the denominator. This is a bit of a mess (for which the author has to admit responsibility), but it works quite well, until in the current paper we encounter quotients of almost typedefinable sets by almost type-definable equivalence relations. Putting these into the existing naming scheme would be complicated, since we would have to say if the "almost" applies to the numerator, denominator, or both. In addition, the prefixes "type-" and "hyper" have somewhat lost of their original meaning, since they not longer designate generalisations of plain first-order definability (we are way past that stage), but merely serve to tell us how to interpret the adverb "almost".We therefore decided that instead of trying to adapt by force a pretty inadequate naming scheme, we should use a new one which would be designed for th...

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Simple Almost Hyperdefinable Groups

Ben-Yaacov

2006

J. Math. Log.

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Fuzzy n-ary subpolygroups

Davvaz

Corsini

Leoreanu-Fotea

2009

Computers & Mathematics with Applications

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Constructing an Almost Hyperdefinable Group

2004

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On the fine structure of the polygroup blow-up

Cited by 5 publications

References 2 publications

Simple Almost Hyperdefinable Groups

Simple Almost Hyperdefinable Groups

Fuzzy n-ary subpolygroups

Constructing an Almost Hyperdefinable Group

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