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<p>We investigate the property of elimination of imaginaries for some special cases of ordered abelian groups. We show that certain Hahn products of ordered abelian groups do not eliminate imaginaries in the pure language of ordered groups. Moreover, we prove that, adding finitely many constants to the language of ordered abelian groups, the theories of the finite lexicographic products <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="153_2025_965_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb {Z}^n\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="153_2025_965_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="52" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb {Z}^n \times \mathbb {Q}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mi>n</mi> </msup> <mo>×</mo> <mi mathvariant="double-struck">Q</mi> </mrow> </math></EquationSource> </InlineEquation> have definable Skolem functions.</p>

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Ordered abelian groups that do not have elimination of imaginaries

  • 期刊论文

  • 发表日期：

    2025年1月15日

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  • Martina Liccardo

摘要

We investigate the property of elimination of imaginaries for some special cases of ordered abelian groups. We show that certain Hahn products of ordered abelian groups do not eliminate imaginaries in the pure language of ordered groups. Moreover, we prove that, adding finitely many constants to the language of ordered abelian groups, the theories of the finite lexicographic products \(\mathbb {Z}^n\) Z n and \(\mathbb {Z}^n \times \mathbb {Q}\) Z n × Q have definable Skolem functions.