The left distributive law and the freeness of an algebra of elementary embeddings

Laver, Richard

doi:10.1016/0001-8708(92)90016-e

Cited by 70 publications

(141 citation statements)

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“…Lemma 2.3 (Dehornoy [1] These results were also proved by Laver [10] under a large cardinal assumption (see below).…”

Section: Free Left-distributive Algebras and Elementary Embeddingsmentioning

confidence: 59%

“…If j is a nontrivial elementary embedding from V λ to V λ , let (A j , ·) and (P j , ·, •) be the subalgebras of (E, ·) and (E, ·, •) generated by j. Laver [10] shows, among other things, that (A j , ·) and (P j , ·, •) are respectively the free monogenic left-distributive algebra and the free monogenic algebra satisfying axioms (LL).…”

Section: Free Left-distributive Algebras and Elementary Embeddingsmentioning

confidence: 99%

“…In this section, we consider algebras of increasing functions from N to N which imitate the behavior of the algebra of elementary embeddings from Laver [10] on ordinals. The existence of such algebras turns out to be equivalent to the properties in Theorem 3.2.…”

Section: Embedding Algebrasmentioning

confidence: 99%

“…The results of Laver [10] show that one can make the set of all elementary embeddings from V λ to itself into a two-sorted embedding algebra by letting O be the set of limit ordinals less than λ and defining ≡ γ to be Laver's γ =. Conversely, many of Laver's arguments about elementary embeddings and their critical points use only the properties of a two-sorted embedding algebra.…”

Section: Definition 42mentioning

confidence: 99%

“…Such algebras became an object of study when they appeared in a set-theoretic context: Laver [10] studied the properties of an algebra of elementary embeddings arising from an extremely large cardinal. He used these properties to prove purely algebraic results about the free left-distributive algebra on one generator; for instance, he showed that the word problem for this algebra is decidable.…”

Section: Introductionmentioning

confidence: 99%

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Left-distributive embedding algebras

Dougherty

Jech

1997

Electron. Res. Announc. Amer. Math. Soc.

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Abstract. We consider algebras with one binary operation · and one generator, satisfying the left distributive law a·(b·c) = (a·b)·(a·c); such algebras have been shown to have surprising connections with set-theoretic large cardinals and with braid groups. One can construct a sequence of finite left-distributive algebras An, and then take a limit to get an infinite left-distributive algebra A∞ on one generator. Results of Laver and Steel assuming a strong large cardinal axiom imply that A∞ is free; it is open whether the freeness of A∞ can be proved without the large cardinal assumption, or even in Peano arithmetic. The main result of this paper is the equivalence of this problem with the existence of a certain left-distributive algebra of increasing functions on natural numbers, called an embedding algebra, which emulates some properties of functions on the large cardinal. Using this and results of the first author, we conclude that the freeness of A∞ is unprovable in primitive recursive arithmetic.

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“…Lemma 2.3 (Dehornoy [1] These results were also proved by Laver [10] under a large cardinal assumption (see below).…”

Section: Free Left-distributive Algebras and Elementary Embeddingsmentioning

confidence: 59%

Section: Free Left-distributive Algebras and Elementary Embeddingsmentioning

confidence: 99%

Section: Embedding Algebrasmentioning

confidence: 99%

Section: Definition 42mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Left-distributive embedding algebras

Dougherty

Jech

1997

Electron. Res. Announc. Amer. Math. Soc.

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On the Algebra of Elementary Embeddings of a Rank into Itself

Laver¹

1995

Advances in Mathematics

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For λ a limit ordinal let E λ Be the set of elementary embeddings j : (V λ , ǫ) → (V λ , ǫ), j not the identity. Then the existence of a λ such that E λ is nonempty is a strong large cardinal axiom (Kanamori-Reinhardt-Solovay [5]). For j ∈ E λ let cr j be the critical point of j. Then λ = sup{j n (cr j) : n < ω} (Kunen [7]).For j, k ∈ E λ let j ·k = α<λ j(k∩V α ); then j ·k ∈ E λ (details are reviewed below).Writing jk for j · k, we have that · is nonassociative, noncommutative, and satisfies the left distributive law a(bc) = (ab)(ac). Letting j • k be the composition of j andimplies the left distributive law (a(bc) = (a • b)c = (ab • a)c = (ab)(ac)). For j ∈ E λ , let A j be the closure of {j} under ·, and let P j be the closure of {j} under · and •. A theorem of [9] is that A j ∼ = A, the free left distributive algebra on one generator, and P j ∼ = P, the free algebra on one generator satisfying .

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Large Ordinals

Jech

1997

Advances in Mathematics

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The left distributive law and the freeness of an algebra of elementary embeddings

Cited by 70 publications

References 5 publications

Left-distributive embedding algebras

Left-distributive embedding algebras

On the Algebra of Elementary Embeddings of a Rank into Itself

Large Ordinals

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