Let be an integer. We show that the set of real numbers that are Poisson generic in base is -complete in the Borel hierarchy of subsets of the real line. Furthermore, the set of real numbers that are Borel normal in base and not Poisson generic in base is complete for the class given by the differences between sets. We also show that the effective versions of these results hold in the effective Borel hierarchy.
We address the question of consistency strength of certain filters and ultrafilters which fail to satisfy the Galvin property. We answer questions [Benhamou and Gitik, Ann. Pure Appl. Logic173 (2022) 103107; Questions 7.8, 7.9], [Benhamou et al., J. Lond. Math. Soc.108(1) (2023) 190–237; Question 5] and improve theorem [Benhamou et al., J. Lond. Math. Soc.108(1) (2023) 190–237; Theorem 2.3].
We present numerous natural algebraic examples without the so-called Canonical Base Property (CBP). We prove that every commutative unitary ring of finite Morley rank without finite-index proper ideals satisfies the CBP if and only if it is a field, a ring of positive characteristic or a finite direct product of these. In addition, we construct a CM-trivial commutative local ring with a finite residue field without the CBP. Furthermore, we also show that finite-dimensional non-associative algebras over an algebraically closed field of characteristic give rise to triangular rings without the CBP. This also applies to Baudisch’s -step nilpotent Lie algebras, which yields the existence of a -step nilpotent group of finite Morley rank whose theory, in the pure language of groups, is CM-trivial and does not satisfy the CBP.
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