We compute the model-theoretic Grothendieck ring, \(K_0({\mathcal {Q}})\), of a dense linear order (DLO) with or without end points, \({\mathcal {Q}}=(Q,<)\), as a structure of the signature \(\{<\}\), and show that it is a quotient of the polynomial ring over \({\mathbb {Z}}\) generated by \({\mathbb {N}}_+\times (Q\sqcup \{-\infty \})\) by an ideal that encodes multiplicative relations of pairs of generators. This ring can be embedded in the polynomial ring over \({\mathbb {Q}}\) generated by \(Q\sqcup \{-\infty \}\). As a corollary we obtain that a DLO satisfies the pigeon hole principle for definable subsets and definable bijections between them—a property that is too strong for many structures.

中文翻译：

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具有密集线性阶的可定义组合

我们计算模型理论上的Grothendieck环\（K_0（{\ mathcal {Q}}）\），具有或不具有端点\（{\ mathcal {Q}} = {Q ，<）\）作为签名\（\ {<\} \）的结构，并表明它是\（{生成的\（{ \ mathbb {Z}} \））上的多项式环的商\ mathbb {N}} _ + \ times（Q \ sqcup \ {-\ infty \}）\），该理想值对生成器对的乘法关系进行编码。该环可以嵌入到\（Q \ sqcup \ {-\ infty \} \）生成的\（{\ mathbb {Q}} \）上的多项式环中。作为推论，我们得出DLO满足鸽子孔原理中可定义的子集和它们之间可定义的双射的要求-对于许多结构而言，此属性太强了。