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On embedding models of arithmetic of cardinality ℵ₁ into reduced powers

Juliette Kennedy, Saharon Shelah (2003)

Fundamenta Mathematicae

In the early 1970’s S. Tennenbaum proved that all countable models of PA₁¯ + ∀₁ -Th(ℕ) are embeddable into the reduced product ℕ ω / ℱ , where ℱ is the cofinite filter. In this paper we show that if M is a model of PA¯ + ∀₁ - Th(ℕ), and |M| = ℵ₁, then M is embeddable into ℕ ω / D , where D is any regular filter on ω.

On the Boffa alternative

B. Bajorska, O. Macedońska (2001)

Colloquium Mathematicae

Let G* denote a nonprincipal ultrapower of a group G. In 1986 M.~Boffa posed a question equivalent to the following one: if G does not satisfy a positive law, does G* contain a free nonabelian subsemigroup? We give the affirmative answer to this question in the large class of groups containing all residually finite and all soluble groups, in fact, all groups considered in traditional textbooks on group theory.

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