排序投射有限群的嵌入性质
The embedding property for sorted profinite groups
AI总结：
该研究在排序投射有限群范畴中引入排序嵌入性质SEP，证明其存在通用SEP覆盖，提出等价的有限排序嵌入性质FSEP并用于证明相关理论的ω-稳定性，还得到通用覆盖的唯一性，推广了经典投射有限群的相关结果。
摘要：
我们研究了排序投射有限群范畴中的嵌入性质。我们引入了排序嵌入性质（SEP）的概念，它与投射有限群的嵌入性质相类似。我们证明了任意排序投射有限群都存在一个通用SEP覆盖。我们的证明为投射有限群通用嵌入覆盖的存在性提供了一种替代证法。此外，我们的证明适用于排序投射有限群的任意满子范畴，只要该子范畴在取有限商、纤维积和逆极限下封闭。\n我们还引入了一个更弱的概念——有限排序嵌入性质（FSEP），并且结果表明它与SEP是等价的。FSEP的优势在于它能够在排序完全系统的一阶语言中被公理化。利用这一点，我们证明了在排序集合为可数的假设下，任意具有SEP的排序投射有限群所对应的排序完全系统的理论是ω-稳定的。在这种情况下，作为副产品，我们得到了排序投射有限群通用SEP覆盖的唯一性，这一结论推广了投射有限群嵌入覆盖的唯一性。
英文摘要：
We study the embedding property in the category of sorted profinite groups. We introduce a notion of the sorted embedding property (SEP), analogous to the embedding property for profinite groups. We show that any sorted profinite group has a universal SEP-cover. Our proof gives an alternative proof for the existence of a universal embedding cover of a profinite group. Also our proof works for any full subcategory of the sorted profinite groups, which is closed under taking finite quotients, fibre product, and inverse limit. We also introduce a weaker notion of finitely sorted embedding property (FSEP), and it turns out to be equivalent to SEP. The advantage of FSEP is to be able to be axiomatized in the first order language of sorted complete systems. Using this, we show that any sorted profinite group having SEP has the sorted complete system whose theory is $ω$-stable under the assumption that the set of sorts is countable. In this case, as a byproduct, we get the uniqueness of a universal SEP-cover of a sorted profinite group, which generalizes the uniqueness of an embedding cover of a profinite group.
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