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The Partial Ordering on the Automorphism Group of the Countable Generic Partial Order

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Abstract

The countable generic poset (P, ≤ ) is the Fraïssé limit of the amalgamation class of finite partially ordered sets (see Glass et al., Math Z 214:55–66, 1993; Schmerl, Algebra Univers 9:317–321, 1979). It is homogeneous and \(\aleph_0\)-categorical with quantifier elimination. This paper concerns the structure (G, ∘ , ≤ ), where \((G,\circ)=\textrm{Aut}(P,\leq)\) and ≤ is the pointwise ordering on G. This is a natural structure to look at, because the ordering on G is ∅-definable up to reversal in the language { ∘ } (but this fact is not proved here). In this paper I show that (G, ≤ ) is elementarily equivalent to (P, ≤ ) itself. More generally, (G, ∘ , ≤ ) satisfies a weakening of the existential closure property for partially ordered groups. (Existential closure in groups has been studied for example in Higman and Scott.) This requires one to study the group G  ∗ , obtained by freely adjoining a finite set of generators to G.

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  1. School of Mathematics, University of Leeds, Leeds, UK

    David Michael Knipe

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  1. David Michael Knipe
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Correspondence to David Michael Knipe.

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Knipe, D.M. The Partial Ordering on the Automorphism Group of the Countable Generic Partial Order. Order 26, 289–307 (2009). https://doi.org/10.1007/s11083-009-9126-8

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  • DOI: https://doi.org/10.1007/s11083-009-9126-8

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