Let ℬ⊆ ℕ∖{1} be a primitive set, ℱℬ := ℤ∖⋃b∈ℬbℤ, η := 1ℱℬ∈{0, 1}ℤ, and denote by Xη the orbit closure of η under the shift. We complement results on heredity of Xη from [Dymek et al., ℬ-free sets and dynamics, Trans. Amer. Math. Soc. 370 (2018) 5425–5489] in two directions: In the proximal case we prove that a certain subshift Xφ ⊇ Xη, which coincides with Xη when ℬ is taut, is always hereditary. (In particular there is no need for the stronger assumption that the set ℬ has light tails, as in [Dymek et al., ℬ-free sets and dynamics, Trans. Amer. Math. Soc. 370 (2018) 5425–5489].) We also generalize the concept of heredity to include the non-proximal (and hence non-hereditary) case by proving that Xφ is always “hereditary above its unique minimal (Toeplitz) subsystem”. Finally, we characterize this Toeplitz subsystem as being a set Xη∗, where η∗ = 1 ℱℬ∗ for a set ℬ∗ that can be derived from ℬ, and draw some further conclusions from this characterization. Throughout results from [Kasjan et al., Dynamics of ℬ-free sets: A view through the window, Int. Math. Res. Not. 2019 (2019) 2690–2734] are heavily used.

中文翻译：

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无ℬ系统中的广义遗传

让ℬ⊆ ℕ∖{1}是一个原始集，ℱℬ ：= ℤ∖⋃b∈ℬbℤ,η ：= 1ℱℬ∈{0, 1}ℤ，并表示为Xη轨道闭合η下班。我们补充了关于遗传的结果Xη来自 [迪梅克等。,ℬ- 自由集和动态，反式。阿米尔。数学。社会党。 370(2018) 5425–5489] 在两个方向上：在最近的情况下，我们证明了某个子移位Xφ ⊇ Xη，这与Xη什么时候ℬ是紧绷的，总是遗传的。（特别是不需要更强的假设，即集合ℬ有轻尾，如 [Dymek等人， ℬ- 自由集和动态，反式。阿米尔。数学。社会党。 370(2018) 5425–5489]。）我们还通过证明将遗传的概念概括为包括非近端（因此非遗传）的情况Xφ总是“遗传于其独特的最小 (Toeplitz) 子系统之上”。最后，我们将这个 Toeplitz 子系统描述为一个集合Xη*， 在哪里η* = 1 ℱℬ*对于一套ℬ*可以从ℬ，并从这个表征中得出一些进一步的结论。[Kasjan] 的整个结果等人., 动力学ℬ- 自由套装：透过窗户看风景，诠释。数学。水库。不是.2019(2019) 2690–2734] 被大量使用。