Every linear order isomorphic to its cube is isomorphic to its square

Ervin, Garrett

doi:10.1016/j.aim.2017.04.010

Cited by 2 publications

(10 citation statements)

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“…Conversely, it can be shown that any order invariant under left multiplication by A 2 is of this form. This is Theorem 3.10 of [4]. We write X = A ω (I [u]2 ).…”

Section: 2mentioning

confidence: 98%

“…Hence the 2-tail-equivalence relation refines the tail-equivalence relation. On the other hand, for a given u ∈ A ω and a ∈ A, observe that for any [4]). We say that such a sequence is eventually periodic, of odd period.…”

Section: 2mentioning

confidence: 99%

“…A fixed-point theorem. The last result we will need for our construction is a fixed-point theorem from [4]. For the remainder of the paper, let A denote the order ω * The appearance of ω * 1 + ω 1 in this theorem is not accidental.…”

Section: 3mentioning

confidence: 99%

“…Do there exist countable non-isomorphic orders X and Y that are righthand divisors of each other? Though analogous questions have since been answered for many other classes of structures, these problems had all remained open until the author solved the first problem negatively in [4]. A negative answer to the fifth problem follows quickly from the work in that paper.…”

Section: Introductionmentioning

confidence: 96%

“…It turns out, however, there is no such order type α: (LO, ×) is one of the rare classes for which the cube property holds, as shown in [4]. It is the unique example, to the author's knowledge, of a natural class of structures for which the cube property holds but Schroeder-Bernstein properties fail.…”

Section: Introductionmentioning

confidence: 99%

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Distinct orders dividing each other on both sides

Ervin

2019

Proc. Amer. Math. Soc.

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We construct non-isomorphic linear orders X and Y that are both left-hand and right-hand divisors of one another, answering positively a question of Sierpiński.2010 Mathematics Subject Classification: 06A05, 03E05. Key words and phrases: linear order, lexicographical product, Schroeder-Bernstein problem 1 2 GARRETT ERVIN Kaplansky test properties. The first of these, sometimes individually referred to as the Schroeder-Bernstein property, asserts that whenever two structures X, Y ∈ K are each isomorphic to a divisor of the other, then X and Y must themselves be isomorphic. That is, ifKaplansky himself was interested in determining whether these properties held for various classes of infinite abelian groups. He considered their failure a heuristic indication that a given class admitted no useful structure theorem, a la the classification theorem for finitely generated abelian groups. In his book [7] in which he recorded them, he said "I believe their defeat is convincing evidence that no reasonable invariants exist." The two properties had been considered previously for other classes of structures by several authors, including Tarski [13] and Hanf [5].A third property, related to the two Kaplansky properties, is the cube property. Whenever a structure X from a class K is isomorphic to its own square X 2 , it is isomorphic to all of its finite powers X n . In particular, it is isomorphic to its cube. The cube property asserts that the converse holds: for all X ∈ K, if X ∼ = X 3 then it is already the case that X ∼ = X 2 . If K contains a counterexample to the cube property, namely, a structure X that is isomorphic to its cube but not to its square, then the pair X and Y = X 2 witnesses the failure of both Kaplansky properties for K.

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“…Conversely, it can be shown that any order invariant under left multiplication by A 2 is of this form. This is Theorem 3.10 of [4]. We write X = A ω (I [u]2 ).…”

Section: 2mentioning

confidence: 98%

Section: 2mentioning

confidence: 99%

Section: 3mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

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