Extensions of two constructions of Ahmad
Extensions of two constructions of Ahmad
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DOI：
10.3233/com-210380
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发表时间：
2022
期刊：
影响因子：
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通讯作者：
Jun Le Goh;S. Lempp;K. Ng;M. Soskova
中科院分区：
文献类型：
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作者：
Jun Le Goh;S. Lempp;K. Ng;M. Soskova
在她1990年的论文中，Ahmad证明了存在所谓的“Ahmad对”，即存在不可比的Σ2-枚举度a0和a1，使得每个枚举度x和a0都是≤a1。同时，她还证明了不存在“对称Ahmad对”，即不存在不可比的Σ2-枚举度a0和a1使得每个枚举度x0和lt；a0都是≤a1，并且使得每个枚举度x1和lt；a1都是≤a0。本文首先给出了Ahmad第二个结果的一个直接证明。然后我们证明了她的第一个结果不能推广到“Ahmad三元组”，即不存在Σ2-枚举度a0，a1和a2使得(a0，a1)和(a1，a2)都是Ahmad对。另一方面，存在一个“弱Ahmad三元组”，即存在两两不可比的Σ2-枚举度a0、a1和a2，使得每个枚举度x<；a0也是≤a1或≤a2；然而，(a0，a1)和(a0，a2)都不是Ahmad对。
In her 1990 thesis, Ahmad showed that there is a so-called “Ahmad pair”, i.e., there are incomparable Σ2-enumeration degrees a0 and a1 such that every enumeration degree x < a0 is ≤ a1. At the same time, she also showed that there is no “symmetric Ahmad pair”, i.e., there are no incomparable Σ2-enumeration degrees a0 and a1 such that every enumeration degree x0 < a0 is ≤ a1 and such that every enumeration degree x1 < a1 is ≤ a0. In this paper, we first present a direct proof of Ahmad’s second result. We then show that her first result cannot be extended to an “Ahmad triple”, i.e., there are no Σ2-enumeration degrees a0, a1 and a2 such that both (a0,a1) and (a1,a2) are an Ahmad pair. On the other hand, there is a “weak Ahmad triple”, i.e., there are pairwise incomparable Σ2-enumeration degrees a0, a1 and a2 such that every enumeration degree x < a0 is also ≤ a1 or ≤ a2; however neither (a0,a1) nor (a0,a2) is an Ahmad pair.