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FINITE LEFT-DISTRIBUTIVE ALGEBRAS

arXiv:math/9209202v1 [math.LO] 8 Sep 1992FINITE LEFT-DISTRIBUTIVE ALGEBRASAND EMBEDDING ALGEBRASRandall Dougherty and Thomas JechOhio State UniversityPennsylvania State UniversityJanuary 27, 1997Abstract. We consider algebras with one binary operation · and one generator (monogenic)and satisfying the left distributive law a · (b · c) = (a · b) · (a · c).

One can define a sequenceof finite left-distributive algebras An, and then take a limit to get an infinite monogenic left-distributive algebra A∞. Results of Laver and Steel assuming a strong large cardinal axiomimply that A∞is free; it is open whether the freeness of A∞can be proved without thelarge cardinal assumption, or even in Peano arithmetic.

The main result of this paper is theequivalence of this problem with the existence of a certain algebra of increasing functions onnatural numbers, called an embedding algebra. Using this and results of the first author, weconclude that the freeness of A∞is unprovable in primitive recursive arithmetic.1.

IntroductionWe consider algebras with one binary operation · and one generator (monogenic) andsatisfying the left distributive law a · (b · c) = (a · b) · (a · c); in particular, we look for arepresentation of the free algebra.The word problem for the free monogenic left-distributive algebra was solved by Laver [6]under the assumption of a large cardinal and subsequently by Dehornoy [4] without suchan assumption. Laver’s result uses elementary embeddings from Vλ into Vλ under the ‘ap-plication’ operation · defined by j ·k = Sα<λ j(k∩Vα).

If there exists such an embedding jother than the identity, then the algebra Aj generated by j is free.When the embeddings in Aj are restricted to an initial segment of Vλ, they form a finitemonogenic left-distributive algebra [7], and these finite algebras can be described withoutreference to elementary embeddings. In fact, for every n there is a (unique) left-distributiveThe first author was supported by NSF grant number DMS-9158092 and by a grant from the Sloanfoundation.The second author was supported by NSF grant number DMS-8918299.Typeset by AMS-TEX1

2RANDALL DOUGHERTY AND THOMAS JECHoperation ∗n on the set A′n = {1, 2, . .

., 2n} such that a ∗n 1 = a + 1 for all a < 2n and2n ∗n 1 = 1.There is a natural way of defining a limit A∞of the algebras A′n, and one can ask whetherA∞is free. We reduce this problem to a simple (Π02) statement of finite combinatorics, andshow that the answer is affirmative provided there exists a nontrivial elementary embeddingfrom Vλ into itself.

The crucial fact used in the proof is a theorem of Laver and Steel [7]on critical points of elementary embeddings.It is open whether the freeness of A∞can be proved without the large cardinal assump-tion, or even in Peano arithmetic. The main result of this paper is the equivalence of thisproblem with the existence of a certain algebra of increasing functions on natural numbers.We introduce embedding algebras, which are algebras (A, ·) of increasing functions a: ω →ω endowed with a binary operation ·.

The axioms for embedding algebras state that theoperation a · b is left distributive and interacts with critical points (the critical point of afunction is the least number moved by the function) in the expected way. If a (nontrivial)embedding algebra A exists, then A∞is free; conversely, we construct an embeddingalgebra under the assumption that A∞is free.The first author proved [5] that the critical sequence for a nontrivial elementary embed-ding j yields an enumeration of critical points in Aj that grows faster than any primitiverecursive function.

One consequence of the main theorem is that such a fast-growing func-tion can be defined under the assumption that A∞is free. It follows that the freenessof A∞is unprovable in primitive recursive arithmetic.2.

The free monogenic left-distributive algebraWe consider algebras with one binary operation · generated by a single generator thatwe denote by the symbol 1. We shall often write ab instead of a · b, and use the conventionthat abc = (ab)c.The left distributive law is the equality(LD)a(bc) = ab(ac).We let W = WA be the set of all words built up from 1 using the operation ·, denote by ≡(or by ≡A) the equivalence relation on W given bya ≡biff(LD) |= a = b,and let A = W/≡be the free left-distributive algebra on one generator.For the rest of this section, let (A, ·) be a left-distributive algebra generated by 1.

Wewill summarize the relevant known results on such algebras.Definition 2.1. We say that a is a left subterm of b, or a

. .

, ck(k > 0), b = ac1 . .

. ck.

FINITE LEFT-DISTRIBUTIVE ALGEBRAS AND EMBEDDING ALGEBRAS3Lemma 2.2. (i) If a

Part (i) is trivial. For (ii), use distributivity: if b = ac1 .

. .

ck, then we have cb =c(ac1 . .

. ck) = ca(cc1) .

. .

(cck).■Theorem 2.3 (Dehornoy [2]). For all a, b ∈A, either a ≡b or a

(This was also proved by Laver [6] under the assumption that

If the relation

If a ̸= b in A,then either a L b, so either πa L πb, so πa ̸= πb; therefore, π isan isomorphism. For left cancellation, if a ̸= b in A, then a

There is an algebra (A, ·) on which

The depth of a ∈W is defined recursively as follows:depth(1) = 0,depth(ab) = max{depth(a), depth(b)} + 1.The herringbone uk of depth k is also defined recursively:u0 = 1,uk+1 = 1uk.One can also define the full word vk, the maximal word of depth k, by v0 = 1 andvk+1 = vkvk. Then vk is equivalent to uk, because an easy induction shows that 1vk = vk+1.Lemma 2.7 (Dehornoy [2, Cor.

2]). If a is a word of depth ≤k, then auk = uk+1 in A.Proof.

By induction on the depth of a (for all k simultaneously).For a = 1, this isimmediate from the definition of uk. If a has positive depth, then a = bc where b and chave depth smaller than that of a, and hence ≤k −1.

Now the induction hypothesis givesabuk = ab(auk−1) = a(buk−1) = auk = uk+1,as desired.■For a ∈W, we write a →LD b when b results from a by a single application of (LD)from left to right (to a subword of a), i.e., replacing x(yz) by xy(xz). We write a →b ifthere is a sequence a0 = a, a1, a2, .

. ., ak = b (k ≥0) such that ai →LD ai+1 for each i < k.

4RANDALL DOUGHERTY AND THOMAS JECHProposition 2.8 (Dehornoy [1]). There is a mapping ∂from W to W with the followingproperties:(1) a →∂a;(2) if a →LD b, then b →∂a;(3) if a →b, then ∂a →∂b.Proof.

First define a binary operation ⊗on W by recursion on the second argument:a ⊗1 = a1,a ⊗bc = (a ⊗b)(a ⊗c). (The effect of a ⊗b is to distribute a in b as many times as possible.

)Then define ∂by another recursion:∂1 = 1,∂(ab) = ∂a ⊗∂b. (The word ∂a contains all possible applications of (LD) within a.

)Now everything used here is (or can be viewed as being) defined by recursion, includ-ing →(in terms of →LD) and even →LD: a →LD b iffeither a has the form a1(a2a3)and b = (a1a2)(a1a3), or a and b have the forms a1a2 and b1b2, respectively, and eithera1 →LD b1 and a2 = b2, or a1 = b1 and a2 →LD b2. One can now prove a sequence ofstatements by straightforward inductions:ab →a ⊗b;(induct on b)a ⊗(b ⊗c) →(a ⊗b) ⊗(a ⊗c);(induct on c)if a →a′, then a ⊗b →a′ ⊗b;(induct on b)if b →LD b′, then a ⊗b →LD a ⊗b′;(induct on b →LD b′)if b →b′, then a ⊗b →a ⊗b′;(induct on b →b′)a →∂a;(induct on a)a1a2(a1a3) →a1 ⊗a2a3;if a →LD b, then b →∂a;(induct on a →LD b)if a →LD b, then ∂a →∂b;(induct on a →LD b)if a →b, then ∂a →∂b.

(induct on a →b)This gives the desired properties.■Lemma 2.9 (Dehornoy [2]). If a

A straightforward induction on the length of the derivation b →b′.■

FINITE LEFT-DISTRIBUTIVE ALGEBRAS AND EMBEDDING ALGEBRAS5Proof of Theorem 2.3. From Proposition 2.8, it follows that, if a ≡A b, then a →∂mb,whenever m is at least the length of an (LD)-derivation of a ≡b.

Now, let a and b bewords in W, and choose k so that both words are of depth ≤k. By Lemma 2.7, we haveauk ≡uk+1 and buk ≡uk+1, so auk →∂muk+1 and buk →∂muk+1 for some m. ByLemma 2.9, there are left subterms a′ and b′ of ∂muk+1 such that a →a′ and b →b′.Since a′ and b′ are left subterms of the same word, we have either a′ = b′, a′

), and it is easy to see that the recursionsare in fact primitive recursions (on the depths of terms, the lengths of derivations, etc. ).Therefore, Theorem 2.3 can be proved in a very basic theory of arithmetic.

One such theoryis Primitive Recursive Arithmetic (PRA), which is formalized in a language containingfunction symbols for all possible function definitions using the constant 0, the successorfunction ′, composition, and primitive recursion; it has axioms stating that the functionsymbols satisfy their definitions, and that 0′ ̸= 0, and a rule of inference allowing inductionon quantifier-free formulas. (See Sieg [8] for more details.

)This theory is among theweakest of the commonly-studied fragments of arithmetic; it is often referred to as theformal version of what Hilbert meant by ‘finitary reasoning.’ It is not hard to show thatthe methods used to prove Theorem 2.3 can be formalized in this theory, so Theorem 2.3is provable in PRA.Now consider algebras with two binary operations · and ◦.We use the conventionab ◦c = (ab) ◦c, a ◦bc = a ◦(bc). Let WP be the set of all words built up from 1 usingboth operations, and let P be the free algebra on one generator under the equivalencea ≡P biff(LL) |= a = b,where (LL) is the following set of axioms (Laver [6]):(LL)a ◦(b ◦c) = (a ◦b) ◦c(a ◦b)c = a(bc)a(b ◦c) = ab ◦aca ◦b = ab ◦aNote that (LD) is a consequence of (LL):a(bc) = (a ◦b)c = (ab ◦a)c = ab(ac).The motivation for axioms (LL) comes from large cardinal theory.

Let Vλ be the col-lection of all sets of rank less than λ, where λ is a limit ordinal. Under the assumptionthat there exists a nontrivial elementary embedding j from Vλ to Vλ, let us consider thealgebra (Aj, ·) generated from j by the operation of applicationj · k =[α<λj(k ∩Vα)

6RANDALL DOUGHERTY AND THOMAS JECHand the algebra (Pj, ·, ◦) generated from j by · and composition of embeddings. Laver [6]shows, among other things, that (Aj, ·) and (Pj, ·, ◦) are respectively the free monogenicleft-distributive algebra and the free monogenic algebra satisfying axioms (LL).Again, we summarize some known facts about the algebras (P, ·, ◦).Let P be an algebra with one generator 1 satisfying (LL).

Let A ⊆P consist of allvalues in P of words in WA; A satisfies (LD) and is generated by 1.Conversely, one can construct an algebra P from an algebra A. The following construc-tion is implicit in Laver [6], and described explicitly in Dehornoy [3, Prop.

2].Proposition 2.10 (Laver, Dehornoy). Any algebra (A, ·) satisfying (LD) can be extendedand expanded to an algebra (P, ·, ◦) satisfying (LL).Proof (sketch).

Given (A, ·), let P ⊇A be the set of formal compositions of one or moreelements of A, with two such formal compositions identified if their equality can be deducedfrom associativity of composition and the rule a ◦b = ab ◦a. Define ◦and · for two suchcompositions a1 ◦· · · ◦an and b1 ◦· · · ◦bm by(a1 ◦· · · ◦an) ◦(b1 ◦· · · ◦bm) = a1 ◦· · · ◦an ◦b1 ◦· · · ◦bm,(a1 ◦· · · ◦an) · (b1 ◦· · · ◦bm) = a1(.

. .

(an(b1)) . .

.) ◦· · · ◦a1(.

. .

(an(bm)) . .

. ).This is well-defined on P and satisfies (LL).■Note that, if A is generated by 1 using ·, then P is generated by 1 using · and ◦.Lemma 2.11.

Every element of the free (LL)-algebra P can be written in the form a1 ◦· · · ◦an for some a1, . .

., an ∈A.Proof. Induct on the form of p as a word in WP.

If p = 1, we are done. Otherwise, p hasthe form qr or q ◦r, where we may assume that q = a1 ◦· · · ◦an and r = b1 ◦· · · ◦bm withai, bj ∈A.

We then haveq ◦r = a1 ◦· · · ◦an ◦b1 ◦· · · ◦bmandqr = c1 ◦· · · ◦cm,where cj = a1(a2(. .

. (an(bj)) .

. .

)), so p has the desired form.■In the following proposition, the left-to-right implication is part of Lemma 3 of Laver [6],while the right-to-left implication uses Lemma 3.2 of that paper.Proposition 2.12. Let (P, ·, ◦) be an algebra satisfying (LL) and generated by 1, and let(A, ·) be the subalgebra of (P, ·) generated by 1.

Then P is free (with respect to (LL)) ifand only if A is free (with respect to (LD)).Proof. First, note that each term a ∈WA is either 1 or of the (unique) form a1b forsome b.

The same statement can be made about b, and so on; we eventually find that

FINITE LEFT-DISTRIBUTIVE ALGEBRAS AND EMBEDDING ALGEBRAS7each such a has a unique expression of the form a1(a2(. .

. (an(1)) .

. .)) for some n ≥0 anda1, .

. ., an ∈WA.The next fact (Laver [6, Lemma 3.2]) we will use is that, if n, m ≥1, ai, bj ∈WA, anda1(a2(.

. .

(an(1)) . .

.)) ≡A b1(b2(.

. .

(bm(1)) . .

. )),thena1 ◦· · · ◦an ≡P b1 ◦· · · ◦bm.It will suffice to show that, if a1(a2(.

. .

(an(1)) . .

.)) →LD b1(b2(.

. .

(bm(1)) . .

. )), thena1 ◦· · · ◦an ≡P b1 ◦· · · ◦bm, since then one can induct on (LD)-derivations.

(Note that anapplication of left distributivity cannot start or finish with the term 1, so no term otherthan 1 is equivalent to 1 under ≡A.) If a1(a2(.

. .

(an(1)) . .

.)) →LD b1(b2(.

. .

(bm(1)) . .

. )),then there are two cases: either the application of left distributivity occurs within a singleterm ai, or it changes ai(ai+1(x)) into aiai+1(ai(x)) for some i.

In the first case, we getfrom a1 ◦· · · ◦an to b1 ◦· · · ◦bm by applying left distributivity within ai; in the secondcase, we get from a1 ◦· · · ◦an to b1 ◦· · · ◦bm by replacing ai ◦ai+1 with aiai+1 ◦ai. Bothof these changes are permitted by (LL), so a1 ◦· · · ◦an ≡P b1 ◦· · · ◦bm.We are now ready to show that, if A is free, then P is free.

Assume A is free, and letp, q ∈WP be words such that p = q in P; we must show that p ≡P q. By Lemma 2.11,there are n, m ≥1 and ai, bj ∈WA such that p ≡P a1 ◦· · · ◦an and q ≡P b1 ◦· · · ◦bm.Since p = q in P, p1 = q1 in P, so (a1 ◦· · · ◦an) · 1 = (b1 ◦· · · ◦bm) · 1 in P, soa1(a2(.

. .

(an(1)) . .

.)) = b1(b2(.

. .

(bm(1)) . .

.)) in P and hence in A.

Since A is free, wehave a1(a2(. .

. (an(1)) .

. .)) ≡A b1(b2(.

. .

(bm(1)) . .

. )).

Now the preceding paragraph givesa1 ◦· · · ◦an ≡P b1 ◦· · · ◦bm, so p ≡P q, as desired.Now assume that P is free; we must show that A is free. To do this, we will show that,if a, b ∈WA and a ̸≡A b, then a ̸= b in A.

By Proposition 2.10, there is an algebra P ′extending the free algebra A which satisfies (LL). Since a ̸≡A b, we have a ̸= b in P ′, soa ̸≡P b.

Since P is free, a ̸= b in P and hence in A. Therefore, A is free.■It is not hard to see that the proof of Propostion 2.12 can be carried out in PRA; onemerely has to use the proof of Proposition 2.10 rather than the proposition itself whenshowing “if a ̸≡A b, then a ̸≡P b.”Now consider the algebras Aj and Pj of elementary embeddings.

For each nontrivialelementary embedding from Vλ to itself, let cr(a) be the critical point of a, the least ordinalmoved by a. Let Γ be the set of all critical points of elements of Aj.

We note thatcr(ab) = a(cr(b)),cr(a ◦b) = min(cr(a), cr(b)).Consequently, the critical point of every a ∈Pj is in Γ, and every a ∈Pj maps Γ into Γ.Theorem 2.13 (Laver and Steel [7]). The set Γ has order type ω.■Theorem 2.14 (Laver [7]).

For every a, b ∈Aj, if a ̸= b, then a(γ) ̸= b(γ) for someγ ∈Γ.■Let κ0 be the critical point of j, and, for all n, let κn+1 = j(κn).

8RANDALL DOUGHERTY AND THOMAS JECHLemma 2.15. (i) If a ∈Aj has depth at most n, then a(κn) = κn+1.

(ii) For every a ∈Pj, there are natural numbers d > 0 and N such that a(κn) = κn+dfor all n ≥N.Proof. (i) By induction on the depth of a:ab(κn) = ab(a(κn−1)) = a(b(κn−1)) = a(κn) = κn+1.

(ii) By Lemma 2.11, we have a = a1 ◦· · · ◦ad for some a1, . .

., ad ∈Aj.■To conclude this section, we remark that one can adjoin to Pj the identity embedding id.The extended algebra still satisfies axioms (LL), as well as these rules:id · a = a,a · id = id,a ◦id = id ◦a = a.3. A sequence of finite algebrasIn this section, we will construct, for each natural number n, an algebra A′n on the set{1, 2, .

. ., 2n} with a binary operation ∗n satisfying the left distributive law.

We will thenconstruct a second operation ◦n on this set so that the resulting two-operation algebra P ′nsatisfies (LL). The subscripts on the operations will sometimes be omitted while a fixed nis being considered.The construction of these algebras is due to Laver; Wehrung proved some additionalproperties of them.

The proof of the following theorem has been reconstructed indepen-dently by several people, including the authors; the presentation here is similar to that ofWehrung [9]. (See also Dehornoy [3, Prop.

7]. )Theorem 3.1′ (mostly Laver).

Let n ≥0. (a) There is a unique left-distributive operation ∗n on {1, 2, .

. ., 2n} such thata ∗n 1 = a + 1 for all a < 2n, and 2n ∗n 1 = 1.

(b) There is a unique additional operation ◦n on {1, 2, . .

., 2n} such that ∗n and ◦nsatisfy axioms (LL).The operation ∗n is defined by double recursion; a∗nb is defined by an outer descendingrecursion on a and an inner ascending recursion on b. The recursive formulas are as follows:(3.1a)2n ∗n b = b;if a < 2n, then(3.1b)a ∗n 1 = a + 1;

FINITE LEFT-DISTRIBUTIVE ALGEBRAS AND EMBEDDING ALGEBRAS9if a < 2n and b < 2n, then(3.1c)a ∗n (b + 1) = (a ∗n b) ∗n (a + 1).In order to see that this is a valid recursion, we must maintain the inductive condition(3.2′)a ∗n b > aif a < 2n.This clearly holds for a ∗n 1. For a ∗n (b + 1) with a < 2n, we have a ∗n b > a by theinduction hypothesis, so (a ∗n b) ∗n (a + 1) has already been defined.

If a ∗n b = 2n, thena∗n (b+1) = 2n ∗n (a+1) = a+1 > a; if a∗n b < 2n, then a∗n (b+1) = (a∗n b)∗n (a+1) >a ∗n b > a. Therefore, (3.2′) holds for a ∗n (b + 1) as well, so the recursion can continue.The equations (3.1) can be deduced from left distributivity and the equations a ∗n 1 =a + 1 (a < 2n) and 2n ∗n 1 = 1.

This is obvious for (3.1b); for (3.1c) and (3.1a), we havea ∗n (b + 1) = a ∗n (b ∗n 1) = (a ∗n b) ∗n (a ∗n 1) = (a ∗n b) ∗n (a + 1),2n ∗n b = 2n ∗n (1 ∗n · · · ∗n 1|{z}b times) = (2n ∗n 1) ∗n · · · ∗n (2n ∗n 1)|{z}b times= 1 ∗n · · · ∗n 1|{z}b times= b.This proves the uniqueness part of Theorem 3.1′(a).An easy induction on b shows that the equations (3.1) hold even when a = 2n, if wetreat addition as being modulo 2n. (Since we are working with the set {1, 2, .

. ., 2n},it will be convenient to treat reduction modulo 2n as a mapping into this set; we willwrite “x mod′ 2n” to mean the unique member of {1, 2, .

. ., 2n} which is congruent to xmodulo 2n.

In particular, 0 mod′ 2n will be 2n.) We will soon show that the equationsalso hold for b = 2n, and prove several other useful properties of A′n at the same time.For any fixed a, consider the sequence a ∗n 1, a ∗n 2, .

. ., a ∗n 2n in A′n.

If a = 2n, thissequence is just 1, 2, . .

., 2n. If a < 2n, then the sequence begins with a+1, and (by (3.1c))each member is obtained from its predecessor by operating on the right by a + 1; hence,by (3.2′), the sequence must be strictly increasing as long as its members remain below 2n.Once 2n is reached (as must happen in at most 2n −a steps), the next member will bea + 1 again, and the sequence repeats.

Therefore, the sequence a ∗n 1, a ∗n 2, . .

., a ∗n 2nis periodic (as long as it lasts); each period is strictly increasing from a + 1 to 2n. We willrefer to the number of terms in each period of this sequence as the period of a in A′n.

(Theperiod of 2n in A′n is 2n. )Proposition 3.2.

(a) The period of any a in A′n is a power of 2; equivalently, a ∗n 2n = 2n for all a. (b) The formulas (3.1) hold modulo 2n in A′n even when a or b is 2n.

(c) Reduction modulo 2n is a homomorphism from A′n+1 to A′n:(a ∗n+1 b) mod′ 2n = (a mod′ 2n) ∗n (b mod′ 2n)for all a, b in A′n+1.

10RANDALL DOUGHERTY AND THOMAS JECH(d) For any a < 2n in A′n, if p is the period of a in A′n, then the period of a+2n in A′n+1is also p, and the period of a in A′n+1 is either p or 2p. The period of 2n in A′n+1 is 2n.Proof.

By simultaneous induction on n. Part (a) for n = 0 is trivial.Suppose (a) holds for n. We noted before that the formulas (3.1) hold modulo 2n whena = 2n. If a < 2n but b = 2n, then (b+1) mod′ 2n = 1 and a∗n 1 = a+1, while a∗n b = 2nby (a), and 2n ∗n (a + 1) = a + 1, so (3.1c) holds even in this case.

Therefore, (b) holdsfor n.Part (c) for n is proved by induction, downward on a and upward on b, as in thedefinition of ∗n+1. If a = 2n+1, then both sides are equal to b mod′ 2n.

If b = 1, then bothsides are equal to (a + 1) mod′ 2n. If a < 2n+1 and b > 1, then the left side is equal to((a mod′ 2n) ∗n ((b −1) mod′ 2n)) ∗n ((a + 1) mod′ 2n)by the induction hypothesis, and the right side is also equal to this value by (b).

Therefore,(c) holds for n.Next, consider (d). Clearly the period of 2n+1 in A′n+1 is 2n+1, twice the period of 2nin A′n.

Now suppose a < 2n, and let p be the period of a in A′n. By (c), for each bin A′n, a ∗n+1 b and (a + 2n) ∗n+1 b must each be equal to either a ∗n b or (a ∗n b) + 2n; ifa ∗n b < 2n, then both of these values are less than 2n+1.

It follows that the periods of aand a + 2n in A′n+1 are at least p. Furthermore, by (3.2′), we must have (a + 2n) ∗n+1 b >a + 2n, so (a + 2n) ∗n+1 b must be equal to (a ∗n b) + 2n for all such b, so, in particular,(a + 2n) ∗n+1 p = 2n+1; hence, the period of a + 2n in A′n+1 is exactly p. (The sameargument shows that the period of 2n in A′n+1 is 2n.) For the period of a in A′n+1, thereare two cases.

If a ∗n+1 p = 2n+1, then the period of a in A′n+1 is p, and we are done. Ifnot, a∗n+1 p must be 2n.

Then a∗n+1 (p+1) must be either a+1 or a+1+2n by (c), andit must be greater than 2n because a ∗n+1 b increases with b until it reaches 2n+1, so wemust have a ∗n+1 (p + 1) = a + 1 + 2n = (a ∗n 1) + 2n. Similarly, using part (c) along with(3.1c) and (3.2′), we see that a ∗n+1 (p + b) = (a ∗n b) + 2n successively for b = 2, 3, .

. ., p.In particular, a ∗n+1 b < 2n+1 for b < 2p and a ∗n+1 2p = 2n+1, so the period of a in A′n+1is 2p.

This completes the proof of (d) for n.Finally, (a) for n + 1 (in the first phrasing) follows immediately from (a) and (d) for n.This completes the induction.■Given these properties of A′n, the proof that the left distributive law holds in A′n is astraightforward triple induction (downward on a and b, upward on c):2n ∗(b ∗c) = b ∗c = (2n ∗b) ∗(2n ∗c);a ∗(2n ∗c) = a ∗c = 2n ∗(a ∗c) = (a ∗2n) ∗(a ∗c);if a, b < 2n, thena ∗(b ∗1) = a ∗(b + 1) = (a ∗b) ∗(a + 1) = (a ∗b) ∗(a ∗1);

FINITE LEFT-DISTRIBUTIVE ALGEBRAS AND EMBEDDING ALGEBRAS11and, furthermore, if c < 2n, thena ∗(b ∗(c + 1)) = a ∗((b ∗c) ∗(b + 1))= (a ∗(b ∗c)) ∗(a ∗(b + 1))[b ∗c > b]= ((a ∗b) ∗(a ∗c)) ∗((a ∗b) ∗(a + 1))= (a ∗b) ∗((a ∗c) ∗(a + 1))[a ∗b > a]= (a ∗b) ∗(a ∗(c + 1)).We now want to define a second operation ◦= ◦n so that the resulting algebraP ′n = ({1, 2, . .

., 2n}, ∗n, ◦n)satisfies Laver’s axioms (LL). In particular, it will have to be true that (a ◦n b) ∗n 1 =a ∗n (b ∗n 1); therefore, we must definea ◦n b = (a ∗n (b + 1)) −1,where the addition and subtraction are performed modulo 2n.

(So we immediately getthe uniqueness in Theorem 3.1′(b).) This definition makes it immediate that reductionmodulo 2n is a homomorphism from P ′n+1 to P ′n.

We now proceed to prove the four laws(LL). All addition and subtraction below is modulo 2n.First, one can show that 2n ◦x = x ◦2n = x as follows:2n ◦x = 2n ∗(x + 1) −1 = (x + 1) −1 = x,x ◦2n = x ∗(2n + 1) −1 = (x ∗1) −1 = x.The proof of (a ◦b) ∗c = a ∗(b ∗c) is by induction on c:(a ◦b) ∗1 = (a ◦b) + 1 = a ∗(b + 1) = a ∗(b ∗1);(a ◦b) ∗(c + 1) = ((a ◦b) ∗c) ∗((a ◦b) + 1) = (a ∗(b ∗c)) ∗(a ∗(b + 1))= a ∗((b ∗c) ∗(b + 1)) = a ∗(b ∗(c + 1)).Next, a ◦b = (a ∗b) ◦a because(a ◦b) + 1 = a ∗(b + 1) = (a ∗b) ∗(a + 1) = ((a ∗b) ◦a) + 1.The proof of the associative law a ◦(b ◦c) = (a ◦b) ◦c is as follows:(a ◦(b ◦c)) + 1 = a ∗((b ◦c) + 1)= a ∗(b ∗(c + 1))= a ∗((b ∗c) ∗(b + 1))= (a ∗(b ∗c)) ∗(a ∗(b + 1))= ((a ◦b) ∗c) ∗((a ◦b) + 1)= (a ◦b) ∗(c + 1)= ((a ◦b) ◦c) + 1.

12RANDALL DOUGHERTY AND THOMAS JECHFinally, to prove that a ∗(b ◦c) = (a ∗b) ◦(a ∗c), proceed by induction downward on b.For b = 2n, we havea ∗(2n ◦c) = a ∗c = 2n ◦(a ∗c) = (a ∗2n) ◦(a ∗c).If b < 2n, thena ∗(b ◦c) = a ∗((b ∗c) ◦b)= (a ∗(b ∗c)) ◦(a ∗b)[b ∗c > b]= ((a ∗b) ∗(a ∗c)) ◦(a ∗b)= (a ∗b) ◦(a ∗c).This completes the proof that P ′n satisfies (LL), so Theorem 3.1′ is proved.The following fact will be useful later:(3.3′)ifa ̸= 2norb ̸= 2n,thena ◦b ̸= 2n.This is proved by cases. If a ̸= 2n, then a ∗(b + 1) > a by (3.2′), so a ∗(b + 1) ̸= 1, soa ◦b ̸= 2n.

If a = 2n but b ̸= 2n, then a ◦b = b ̸= 2n.We remark that Theorem 3.1′ can be rephrased slightly, replacing 2n by 0:Theorem 3.1 (same credits as for 3.1′). There are unique operations ∗n and ◦n on An =Pn = {0, 1, .

. ., 2n −1} such that the axioms (LL) hold and, for all a ∈Pn,a ∗n 1 = a + 1 mod 2n.■This has no effect on the structure of the algebras, but it affects statements referring tothe ordering of the elements of the algebra.

In particular, (3.2′) and (3.3′) become:eithera ∗n b = 0ora ∗n b > a;(3.2)ifa ̸= 0orb ̸= 0,thena ◦b ̸= 0. (3.3)Also, the ordinary mod operation now gives the homomorphism from Pn+1 to Pn.The element 0 (or 2n) of the algebra plays the role that the identity embedding playedat the end of section 2:0 ∗a = a,a ∗0 = 0,a ◦0 = 0 ◦a = a.

FINITE LEFT-DISTRIBUTIVE ALGEBRAS AND EMBEDDING ALGEBRAS134. The limit algebras A∞and P∞Using the finite algebras (An, ∗n) and (Pn, ∗n, ◦n), we construct monogenic algebras(A∞, ·) and (P∞, ·, ◦).

Let WA ⊂WP be the sets of words built up from 1 using · and us-ing ·, ◦, respectively. The set of all positive integers can be embedded in WA by identifyingeach positive integer with a word in WA, by recursion:1 = 1,a + 1 = a · 1.We also adjoin 0 to WP, letting W ∗P = WP ∪{0} and W ∗A = WA ∪{0}, and add rules0 · a = a,a · 0 = 0,a ◦0 = 0 ◦a = a.For every word a ∈W ∗P and every n ≥0, let [a]n be the value of a in Pn = {0, 1, .

. ., 2n−1},and consider the equivalence relation ≡∞defined by:a ≡∞biff[a]n = [b]n for all n ≥0.We let A∞and P∞be, respectively, the quotients by ≡∞of WA and WP.

Clearly A∞and P∞are generated by 1; also, they satisfy (LD) and (LL), respectively, because Anand Pn do. (In fact, an equivalent definition for A∞and P∞is that they are the subalgebrasgenerated by 1 of the inverse limits of the algebras An and Pn, respectively.) Moreover,A∞⊆P∞.

We shall investigate the possibility that A∞or P∞is free.Lemma 4.1. For every a ∈WP and every n, [a]n+1 is either [a]n or [a]n + 2n.Proof.

This follows immediately from the fact that reduction modulo 2n is a homomor-phism from Pn+1 to Pn.■Note that, as a corollary, if [a]n ̸= 0, then [a]n+1 ̸= 0.Definition 4.2. Let a ∈WP be such that [a]n ̸= 0 for some n. The signature s(a) of a isthe largest n such that [a]n = 0.By Lemma 4.1, for each n > s(a), 2s(a) is the largest power of 2 which divides [a]n.Lemma 4.3.

Let a, b ∈WP be such that [b]n ̸= 0 for some n. Then, for every n ≥0,[ab]n = 0iff[a · 2s(b)]n = 0.Proof. If [a · 2s(b)]n = 0, then [a]n ∗n [2s(b)]n = 0, so 2s(b) is a multiple of the period of [a]nin Pn.

But [b]n is a multiple of 2s(b), so [a]n ∗n [b]n = 0, so [ab]n = 0.On the other hand, suppose [ab]n = 0; then [a]n∗n[b]n = 0. If s(b) ≥n, then [2s(b)]n = 0,so [a]n ∗n [2s(b)]n = 0.

If s(b) < n, let q be the period of [a]n in An; then q divides [b]n,and since q is a power of 2, q divides the largest power of 2 dividing [b]n, which is 2s(b).This again gives [a]n ∗n [2s(b)]n = 0. Hence, in either case, [a · 2s(b)]n = 0.■

14RANDALL DOUGHERTY AND THOMAS JECHCorollary. s(ab) = s(a · 2s(b)).Theorem 4.4.

The following are equivalent:(i) (A∞, ·) is free. (ii) (P∞, ·, ◦) is free.

(iii) A∞satisfies the left cancellation law. (iv)

(v) If a

(viii) For every k ≥1, there is an n such that [1 · k]n ̸= 0.Proof. (i)↔(ii): Proposition 2.12.

(i)→(viii): Assume that, for all n, [1 · k]n = 0. Then, in each An, 1 ∗(k + 1) = (1 ∗k) ∗2 =0 ∗2 = 2 = 1 ∗1.

However, it is easy to see that the word 1 · 1 is inequivalent inthe free algebra to any other word, because no application of the distributive lawcan start from or result in 1 · 1. Therefore, A is not free.

(viii)→(vii): By induction on k ≥0, we prove that [uk]n ̸= 0 for some n. Assume that this istrue for k, and let s = s(uk) be the signature of uk. Let n be such that [1·2s]n ̸= 0.By Lemma 4.3, we have [uk+1]n = [1 · uk]n ̸= 0.

(vii)→(vi): Let k be the depth of a. We show that, if [a]n = 0, then [u]n = 0, where u = uk.By Lemma 2.7, au = uk+1 = uu.

If [a]n = 0, then [au]n = 0 ∗[u]n = [u]n; since[u]n ∗[u]n is either 0 or >[u]n by formula (3.2), we have [u]n = 0. (vi)→(v): If [a]m ̸= 0 for some m, then [a]n ̸= 0 for all n ≥m.Suppose a

. .

ck. Let n be sufficiently large that [a]n ̸= 0, [ac1]n ̸= 0, [ac1c2]n ̸=0, .

. ., [b]n ̸= 0.

By (3.2), we have [a]n < [ac1]n < · · · < [b]n.(v)→(iv): Trivial. (iv)→(iii): Lemma 2.4.

(iii)→(viii): As for (i)→(viii), if [1 · k]n = 0 for all n, then 1 · (k + 1) = 1 · 1 in A∞, violatingleft cancellation. (iv)→(i): Lemma 2.4.■All of the steps here can be formalized in primitive recursive arithmetic, so Theorem 4.4is a theorem of PRA.5.

Embedding algebrasIn this section, we consider algebras of increasing functions from ω to ω which imitatethe behavior of the algebra of elementary embeddings from Laver [6] when restricted tothe set of critical points. The existence of such algebras will turn out to be equivalent tothe properties in Theorem 4.4.

Moreover, this equivalence can be proved (and formulated)in primitive recursive arithmetic.

FINITE LEFT-DISTRIBUTIVE ALGEBRAS AND EMBEDDING ALGEBRAS15Let id be the identity function on ω. If f: ω →ω is strictly increasing and differentfrom id, let cr(f) be the least n such that f(n) > n (the critical point of f).Definition 5.1.

An embedding algebra is a structure (A, ·) where A is a collection ofstrictly increasing functions from ω to ω, · is a left-distributive binary operation on A,and, for every a, b ∈A with b ̸= id, cr(a · b) = a(cr(b)).As usual, we will often write ab instead of a · b. The set A need not contain the identityfunction, but, if it does not, one can extend the operation · to A∪{id} in the obvious way:a · id = id, id · a = a.An embedding algebra A is nontrivial if it has an element other than id.

Note thatthe set of non-identity elements of A is closed under ·: if b has a critical point, so doesa · b. Also, if A is nontrivial, then A has infinitely many critical points: if n = cr(a), thena(n) = cr(aa) and a(n) > n.The main goal of the next three sections will be to prove the following theorem.Theorem 5.2.

The statement “There exists a nontrivial embedding algebra” is equivalentto the statement “A∞is free”.When proving that “A∞is free” implies the existence of an embedding algebra, we shallsee that there is a natural way of associating increasing functions from ω to ω with wordsin WA. However, it is not easy to prove that inequivalent words yield distinct functions.Here we shall rely on Theorem 2.14, but first we have to develop techniques to ‘miniaturize’Laver’s proof.

This will be done in Section 6. In order to develop the necessary machinery,we first define a different kind of ‘embedding algebra.’ The new definition will includemuch of Laver’s machinery explicitly; the resulting structure will be much less concretebut more amenable to algebraic manipulation.Definition 5.3.

A two-sorted embedding algebra consists of a nonempty set E (the ‘em-beddings,’ for which we will use variables a, b, . .

.) and a nonempty set O (the ‘ordinals,’for which we will use variables α, β, .

. .

), together with binary operations · and ◦on E, abinary relation ≤on O, a constant id ∈E, an application operation a, β 7→a(β) (whichwill often be written without parentheses) from E × O to O, a function cr: E−{id} →O,and a ternary relation ≡⊆E × O × E, satisfying the following axioms:• The relation ≤is a linear ordering of O.• Embeddings are strictly increasing monotone functions:β < γimpliesaβ < aγ,andaβ ≥β.• For all a ̸= id, a(cr(a)) > cr(a).• The operation ◦represents composition: (a ◦b)γ = a(bγ).• The constant id represents the identity:id(γ) = γ,a · id = id,andid · a = a ◦id = id ◦a = a.• The axioms (LL) hold.

16RANDALL DOUGHERTY AND THOMAS JECH• For each γ, ≡γ is an equivalence relation on E which respects · and ◦(i.e., if a ≡γ a′and b ≡γ b′, then a · b ≡γ a′ · b′ and a ◦b ≡γ a′ ◦b′).• If γ ≤δ and a ≡δ b, then a ≡γ b.• If a ≡γ b and aδ < γ, then aδ = bδ.• For any a ̸= id, a ≡cr(a) id.• Coherence: a ≡γ b implies ca ≡cγ cb.It follows from these axioms that the operation · distributes over itself and application:a(bc) = ab(ac)anda(bγ) = ab(aγ).A few more properties also follow easily:Proposition 5.4. In a two-sorted embedding algebra, if a and b are embeddings differentfrom id, then:(1) cr(a) is the least ordinal moved by a;(2) cr(ab) = a(cr(b)); and(3) cr(a ◦b) = min(cr(a), cr(b)).Proof.

It is given that a(cr(a)) > cr(a); if β < cr(a), then the fact that id ≡cr(a) a impliesthat β = id(β) = a(β).Since id ≡cr(b) b, coherence gives id = a · id ≡a(cr(b)) ab, soab does not move any ordinal less than a(cr(b)); but it moves a(cr(b)) to ab(a(cr(b))) =a(b(cr(b))) > a(cr(b)), so we must have cr(ab) = a(cr(b)). For (3), let γ = min(cr(a), cr(b)).Then, since ≡γ respects ◦, we have id ≡γ a ◦b, while (a ◦b)γ = a(bγ) ≥max(aγ, bγ) > γ,so γ is the least ordinal moved by a ◦b.■It is easy to verify that all of the axioms in Definition 5.3 are preserved when onemoves to a substructure (replacing E and O with smaller sets closed under the operations,and restricting the operations and relations accordingly).

In particular, if one keeps thesame E but replaces O with the range of the function cr (assuming that E ̸= {id}), thenProposition 5.4(2) implies that the new sets are closed under the operations, so one obtainsa new two-sorted embedding algebra in which every ordinal is a critical point.If desired, one can restrict E to the embeddings obtained from a single embedding j ̸= idusing · and ◦, along with id; this gives a two-sorted embedding algebra generated by a singleembedding. From now on, we will call a two-sorted embedding algebra monogenic if itsnon-identity embeddings are generated from a single non-identity embedding via · and ◦.Similarly, an embedding algebra is monogenic if it is generated from a single non-identityembedding via ·; any nontrivial embedding algebra has monogenic subalgebras.

Note thata monogenic embedding algebra does not contain the identity function.The results of Laver [6] show that one can make the set of all elementary embeddingsfrom Vλ to itself into a two-sorted embedding algebra by letting O be the set of limitordinals less than λ and defining ≡γ to beγ= (as defined in Laver [6], Section 2). We nowwant to show that just the simple properties of embedding algebras suffice to constructthe more elaborate apparatus of a two-sorted embedding algebra.

FINITE LEFT-DISTRIBUTIVE ALGEBRAS AND EMBEDDING ALGEBRAS17Proposition 5.5. If a nontrivial embedding algebra exists, then there exists a two-sortedembedding algebra in which the ordinals have order type ω.Proof.

Let such an embedding algebra be given; we will construct a two-sorted embeddingalgebra. The ordinal set O will be the set of critical points from the given algebra; this isan infinite subset of ω, so it has order type ω.

The embedding set and the operations andrelations will be built up in several steps.To start with, let E1 be the set of non-identity embeddings in the given algebra. Asnoted before, this set is closed under ·.

Now the following properties are true of E1 and O:• The left distributive law holds.• β < γ implies aβ < aγ.• a(γ) ≥γ.• a(cr(a)) > cr(a).• a(γ) = γ for γ < cr(a).• cr(ab) = a(cr(b)).We also have the property• ab(aγ) = a(bγ),since every ordinal γ is a critical point andab(a(cr(c))) = ab(cr(ac)) = cr(ab(ac)) = cr(a(bc)) = a(cr(bc)) = a(b(cr(c))).Now use the construction from Proposition 2.10 to extend and expand (E1, ·) to analgebra (E2, ·, ◦) satisfying Laver’s laws (LL). The application operation on these newembeddings is defined naturally: each embedding a is a formal composition (a1 ◦· · · ◦an)of members of E1, and we let a(γ) = a1(a2(.

. .an(γ) .

. .

)).We have aiai+1(ai(δ)) =ai(ai+1(δ)) for any δ, so replacing ai ◦ai+1 with aiai+1 ◦ai in the formal composition doesnot change the resulting value of a(γ); since formal compositions were identified only whenone could transform one into the other by such replacements and/or the reverse, the valuea(γ) is well-defined. Also, let cr(a) be the minimum of cr(a1), .

. ., cr(an); this is the least γsuch that a(γ) > γ, so it also does not depend on the expression for a.

Then we have:• (LL) holds.• (a ◦b)γ = a(bγ).• cr(a ◦b) = min(cr(a), cr(b)).And the properties listed before hold for E2 as well.Let E be E2 ∪{id}, where id is a new embedding for which cr(id) is not defined but theother operations are defined by:• id(γ) = γ, a · id = id, and id · a = a ◦id = id ◦a = a.Again the previous properties continue to hold. Now it only remains to define a ≡γ b sothat the rest of the axioms in Definition 5.3 hold.Lemma 5.6.

Assume the facts listed above. Let a, b1, .

. ., bk be embeddings, where k ≥0,and let γ be an ordinal.

18RANDALL DOUGHERTY AND THOMAS JECH(i) If cr(a) > b1b2 · · · bkγ, then ab1b2 · · · bkγ = b1b2 · · · bkγ. (ii) If cr(a) > ab1b2 · · · bkγ, then ab1b2 · · · bkγ = b1b2 · · · bkγ.Proof.

These are both proved by induction on k (simultaneously for all embeddings). Letus write (im) for the case k = m of (i), and similarly for (ii).

Note that the hypotheses of(i) and (ii) each imply that cr(a) > γ. (i0): This just says that a does not move any ordinal below its critical point.

(i1): ab1γ = ab1(aγ) = a(b1γ) = b1γ. (ik) for k ≥2: Let s = ab1a.

Note that s(ab1b2) = ab1a(ab1b2) = ab1(ab2) = a(b1b2).Also note that cr(s) = ab1(cr(a)) ≥cr(a); similarly, cr(ws) ≥cr(a) for any w. In particular,cr(b1b2 · · · bk−1s) ≥cr(a) > b1b2 · · · bkγ,so (i1) givesb1b2 · · · bk−1(sbk)γ = b1b2 · · · bk−1s(b1b2 · · ·bk−1bk)γ = b1b2 · · · bk−1bkγ.We now havecr(b1b2 · · · bk−2s) ≥cr(a) > b1b2 · · · bk−1(sbk)γ,so, if k > 2, we can apply (i2) to getb1b2 · · · bk−2(sbk−1)(sbk)γ = b1b2 · · · bk−2s(b1b2 · · ·bk−2bk−1)(sbk)γ= b1b2 · · · bk−1(sbk)γ = b1b2 · · · bk−1bkγ.We can now apply (i3) to b1b2 · · · bk−3s, and so on all the way to (ik−2), to getb1b2(sb3)(sb4) · · ·(sbk)γ = b1b2 · · · bkγ.Now we haves(ab1b2 · · · bkγ) = s(ab1b2)(sb3)(sb4) · · · (sbk)(sγ)= a(b1b2)(sb3)(sb4) · · ·(sbk)γ= b1b2(sb3)(sb4) · · ·(sbk)γby (ik−1)= b1b2 · · ·bkγ= s(b1b2 · · · bkγ).Since s maps distinct ordinals to distinct ordinals, we get ab1b2 · · · bkγ = b1b2 · · · bkγ. (ii0): We have cr(a) > aγ ≥γ, so (i0) applies.

(ii1): ab1γ = ab1(aγ) = a(b1γ) ≥b1γ, so cr(a) > b1γ, so (i1) applies. (iik) for k ≥2: Again let s = ab1a.

We now havecr(ws) ≥cr(s) ≥cr(a) > ab1b2 · · · bkγ

FINITE LEFT-DISTRIBUTIVE ALGEBRAS AND EMBEDDING ALGEBRAS19for any w. This givesab1b2 · · · bkγ = s(ab1b2 · · · bkγ)= s(ab1b2)(sb3)(sb4) · · ·(sbk)(sγ)= a(b1b2)(sb3)(sb4) · · · (sbk)γ= b1b2(sb3)(sb4) · · ·(sbk)γby (iik−1)= b1b2s(b1b2b3)(sb4) · · ·(sbk)γ= b1b2b3(sb4) · · ·(sbk)γby (iik−2)= b1b2b3s(b1b2b3b4)(sb5) · · ·(sbk)γ= b1b2b3b4(sb5) · · · (sbk)γby (iik−3)= · · ·= b1b2 · · · bkγ,by (ii1)as desired.■Define the preliminary relation ≃γ between embeddings as follows: a ≃γ b if, for eachk ≥0 and all embeddings c1, . .

., ck,ac1 · · · ck ↿γ = bc1 · · · ck ↿γ,where a ↿γ is a ↾{β: a(β) < γ}. In other words, a ≃γ b iff, for any δ, if either ac1 · · · ckδor bc1 · · · ckδ is less than γ, then ac1 · · · ckδ = bc1 · · ·ckδ.

This is easily seen to be anequivalence relation, and Lemma 5.6 just states that a ≃cr(a) id.We can now define the final desired relation ≡γ by: a ≡γ b iffra ≃rγ rb for allembeddings r (including r = id). This is also an equivalence relation.

Since cr(ra) =r(cr(a)), we have a ≡cr(a) id.If a ≡γ b, then (r ◦c)a ≃(r◦c)γ (r ◦c)b for any r, so r(ca) ≃r(cγ) r(cb); hence, ca ≡cγ cb.Easily, if γ ≤δ, then a ≃δ b implies a ≃γ b, and the same holds for ≡.It follows immediately from the definitions of ≡γ (with r = id) and ≃γ (with k = 0)that, if a ≡γ b and aδ < γ, then aδ = bδ.If a ≡γ a′ and b ≡γ b′, then we have already shown that ab ≡aγ ab′, so ab ≡γ ab′. Also,r(ab)c1 · · · ck = ra(rb)c1 · · · ck and r(a′b)c1 · · · ck = ra′(rb)c1 · · · ck, so from a ≡γ a′ we getab ≡γ a′b.

Similarly, we get (a ◦b) ≡γ (a′ ◦b) since r(a ◦b)c1 · · · ck = ra(rbc1)c2 · · · ck andthe same for a′. (For the case k = 0, note that, if r(a ◦b)δ < rγ, then ra(rbδ) < rγ, sora(rbδ) = ra′(rbδ), so r(a ◦b)δ = r(a′ ◦b)δ.) Now, using the formulas a ◦b = ab ◦a anda◦b′ = ab′ ◦a, we get (a◦b) ≡γ (a◦b′).

So the equivalence relation ≡γ respects applicationand composition of embeddings.Therefore, we have a two-sorted embedding algebra.■If the original embedding algebra satisfies the property ab(a(n)) = a(b(n)) for all em-beddings a, b and natural numbers n, then one can let O be the entire set ω, rather thanjust the critical points, and the construction will work as before. As a result, one sees that

20RANDALL DOUGHERTY AND THOMAS JECHthe two-sorted embedding algebra includes an ‘isomorphic’ copy of the original embeddingalgebra, expressed in two-sorted form. [In order to see that deleting id and reinserting itlater does not cause a problem, we must show that the new formulas for multiplying by idmatch the old ones.

In other words, we must see that, if the original embedding algebracontained id, then it satisfied id · a = a and a · id = id. To see this, use the property aboveto get, for all n,(id · a)(n) = (id · a)(id(n)) = id(a(n)) = a(n)and(a · id)(a(n)) = a(id(n)) = a(n) = id(a(n)).So id· a = a, and a · id agrees with id at all numbers of the form a(n); but the only strictlyincreasing function from ω to ω which agrees with id at infinitely many places is id.

]It is easy to see that, if the original embedding algebra is monogenic, then so is thetwo-sorted embedding algebra constructed above.We conclude this section with a proposition about two-sorted embedding algebras whichis a substitute for Kunen’s theorem about elementary embeddings.For any non-identity embedding a, the sequence cr(a), a(cr(a)), a(a(cr(a))), . .

. is astrictly increasing sequence of ordinals, called the critical sequence of a.Proposition 5.7.

In any monogenic two-sorted embedding algebra, if a ̸= id is an embed-ding, then the critical sequence of a is cofinal in the set of critical points (the range of cr).Also, a(γ) > γ for any critical point γ ≥cr(a).Proof. All members of the critical sequence are critical points (of the embeddings a, aa,a(aa), a(a(aa)), etc.).

Let j be a non-identity embedding which generates the algebra,and let ⟨κn: n ∈ω⟩be the critical sequence of j. We recall Lemma 2.15.

It was stated forelementary embeddings, but the proof clearly works in the present context as well. Thusevery a must move some ordinal κn, and hence cr(a) ≤κn; this shows that the criticalsequence of j is cofinal in the critical points.

To complete the proof of the first claim, wenow show by induction on expressions in j that, if a ̸= id and ⟨αn: n ∈ω⟩is the criticalsequence of a, then αn ≥κn for all n. This is again trivial for a = j. Suppose it is true forb and c, with critical sequences ⟨βn: n ∈ω⟩and ⟨γn: n ∈ω⟩respectively.

If a = bc, theninduction gives αn = bγn for all n, so αn = bγn ≥γn ≥κn. If a = b ◦c, then α0 is eitherβ0 or γ0.

In the former case, the fact that αn+1 = b(cαn) ≥bαn gives αn ≥βn for all n;similarly, in the latter case, we have αn ≥γn for all n. In either case, we get αn ≥κn, asdesired.Now, if γ ≥cr(a) is a critical point, then γ ≥α0 and γ < αm for some m, so there isan n such that αn ≤γ < αn+1. This gives aγ ≥aαn = αn+1 > γ.■6.

Extended two-sorted embedding algebrasIn order to prove Theorem 5.2, we will need to perform a number of the argumentsof Laver [7] in the context of two-sorted embedding algebras. This is straightforward forarguments involving only the operations which are built into these algebras, but some ar-guments use additional features of elementary embeddings.

In particular, a few arguments

FINITE LEFT-DISTRIBUTIVE ALGEBRAS AND EMBEDDING ALGEBRAS21use ordinals of the form a(<γ), defined to be the least ordinal greater than a(β) for allβ < γ. In this section, we will define an extended algebra which includes this operation andshow that such algebras can be constructed from ordinary two-sorted embedding algebras;this will allow us to use this new operation to prove facts about the original algebra.Definition 6.1.

An extended two-sorted embedding algebra is a two-sorted embeddingalgebra (with embedding set E and ordinal set O), together with two new operations, acofinality function cf: O →O and a mapping from E ×O to O for which we use the notationa, γ 7→a(<γ), satisfying the following additional axioms:a(b(<γ)) = ab(

In an extended two-sorted embedding algebra:(1) a(<γ) ≥γ;(2) id(<γ) = γ;(3) a(<γ) = γ for γ ≤cr(a); and(4) if a(cf γ) > cf γ, then a(<γ) < aγ.Proof. For all δ < γ, we have a(<γ) > aδ ≥δ; hence, (1) holds.

This and id(<γ) ≤id(γ)give (2); we then get (3) because a ≡γ id. For (4), we have a(<γ) ≤aγ, and equalitycannot hold because a(<γ) and aγ have different cofinalities.■Again it is not hard to verify that the axioms for an extended two-sorted embeddingalgebra hold in the case where E is a set of elementary embeddings on Vλ and O is thecollection of limit ordinals less than λ [7].Also, any subalgebra of an extended two-sorted embedding algebra is also an extended two-sorted embedding algebra; in particular,if we keep the same set of embeddings but restrict the ordinals to those of the forma(

(Proposition 6.2(3) gives

22RANDALL DOUGHERTY AND THOMAS JECHcr(a) = a(

Suppose that we are given a two-sorted embedding algebra, in which everyordinal is a critical point. Then the algebra can be extended to a new two-sorted embeddingalgebra with the same embedding set, on which the required additional operations can bedefined so as to give an extended two-sorted embedding algebra.The proof of this theorem will use the following two lemmas about two-sorted embeddingalgebras.Lemma 6.4.

In any two-sorted embedding algebra, if γ = cr(c), then:(a) cc(caγ) < c(caγ);(b) caγ is not in the range of c.Proof. For (a), note that cr(cc) = cγ > γ, so ccγ = γ; hence,c(caγ) = cc(ca)(cγ) > cc(ca)γ = cc(ca)(ccγ) = cc(caγ).On the other hand, an element δ of the range of c cannot satisfy ccδ < cδ; if δ = cβ, thencδ = cc(cβ) = ccδ.

Therefore, (b) holds.■Lemma 6.5. In any two-sorted embedding algebra, if cr(r) = cr(s) = κ, then rλ < raκimplies sλ < saκ.Proof.

Assume rλ < raκ. Note that cr(rs) = rκ > κ, so rsκ = κ; this givesr(sλ) = rs(rλ) < rs(raκ) = rs(ra)(rsκ) = rs(ra)κ < rs(ra)(rκ) = r(saκ).Since r gives an increasing function on the ordinals, we must have sλ < saκ.■Proof of Theorem 6.3.

Fix a two-sorted embedding algebra. Let E and O be its embeddingset and ordinal set, respectively, and assume that the range of cr is all of O.

We mustextend O to a larger collection of ordinals on which the operation a(<γ) can be suitablydefined. The remarks following Proposition 6.2 indicate that this new set of ordinals needonly contain the ordinals a(

In an extended two-sorted embedding algebra, if γ = cr(c) and δ is anyordinal, thena(<γ) ≤δ ⇐⇒caγ < cδ.Proof. If a(<γ) ≤δ, then the fact that cγ > γ givescaγ < ca(

FINITE LEFT-DISTRIBUTIVE ALGEBRAS AND EMBEDDING ALGEBRAS23On the other hand, if δ < a(<γ), then we can use c(<γ) = γ to getcδ < c(

If c′ also has critical point γ, then Lemma 6.5 givesc(d ◦a)γ ≤c(dbδ) ⇐⇒c′(d ◦a)γ ≤c′(dbδ),so cd(caγ) ≤c(dbδ) iffc′d(c′aγ) ≤c′(dbδ). Also, if d′ is another embedding with criticalpoint δ, then cr(cd) = cr(cd′) = cδ, so Lemma 6.5 givescd(caγ) < cd(cb)(cδ) ⇐⇒cd′(caγ) < cd′(cb)(cδ).Note that, by Lemma 6.4(b), c(d ◦a)γ ≤c(dbδ) is equivalent to c(d ◦a)γ < c(dbδ), so(a, γ)R(b, δ) iffcd(caγ) < c(dbδ).

Therefore, R is well-defined.Lemma 6.4(a) implies that R is reflexive. We will now show that R is transitive.

Suppose(a, ρ)R(b, σ) and (b, σ)R(c, τ); fix embeddings r, s, t with critical points ρ, σ, τ, respectively.We then have rs(raρ) ≤r(sbσ) and st(sbσ) ≤s(tcτ), sors(rt(raρ)) = rs(rt)(rs(raρ))≤rs(rt)(r(sbσ))= r(st(sbσ))≤r(s(tcτ))= rs(r(tcτ)),so rt(raρ) ≤r(tcτ), so (a, ρ)R(c, τ). The same proof using > instead of ≤shows that thenegation of R is also transitive.

24RANDALL DOUGHERTY AND THOMAS JECHWe now know that R is a preorder; if we define the relation ∼on E × O by(a, γ) ∼(b, δ) ⇐⇒(a, γ)R(b, δ) and (b, δ)R(a, γ),then ∼is an equivalence relation on E × O and R induces a partial order on the set ofequivalence classes. Let O∗be the set of equivalence classes; we will write [a, γ] for theequivalence class of (a, γ).

Let ≤∗be the partial ordering induced by R on O∗. We thenhave[a, γ] ≤∗[b, δ] ⇐⇒cd(caγ) ≤c(dbδ)⇐⇒cd(caγ) < c(dbδ),where cr(c) = γ and cr(d) = δ.

The fact that the negation of R is transitive implies thatany two elements of E × O are R-comparable (if xRy and yRx were both false, then xRxwould be false, contradicting reflexivity), so ≤∗is a linear ordering of O∗.The various distributive laws imply that, for any e ∈E, we have (a, γ)R(b, δ) if andonly if (ea, eγ)R(eb, eδ). Therefore, e induces a mapping from O∗to O∗via the formulae[a, γ] = [ea, eγ], and this mapping is strictly increasing.

Also, we clearly have (e◦e′)[a, γ] =e(e′[a, γ]).The element [a, γ] of O∗is meant to represent a(<γ) in an extended algebra. For this toextend the original algebra, we need an element H(γ) of O∗to correspond to each γ ∈O.This element will turn out to be [c, γ], where c is any embedding with critical point γ. Inorder to see that this is well-defined and gives the proper ordering on the representativesin O∗, we need the following result.Lemma 6.7.

If cr(c) = γ and cr(d) = δ, then [c, γ] ≤∗[d, δ] if and only if γ ≤δ.Proof. By definition, [c, γ] ≤∗[d, δ] if and only if cd(ccγ) ≤c(ddδ).

But cr(cc) > γ andcr(dd) > δ, so this is equivalent to cdγ ≤cδ. Now, if γ = δ, then cr(cd) = cγ > γ, socdγ = γ = δ ≤cδ.

If γ > δ, then cdγ ≥γ > δ = cδ, so [c, γ] ̸≤∗[d, δ], so [c, γ] >∗[d, δ].Symmetrically, if γ < δ, then [c, γ] <∗[d, δ].■So the correspondence between γ and [c, γ] gives an order-preserving map H: O →O∗.This lets us define the new critical point map cr∗: E →O∗by the formula cr∗(c) =H(cr(c)) = [c, cr(c)].We next verify that the embedding maps γ∗7→eγ∗satisfy e[a, γ] ≥∗[a, γ]. We mustshow that c(ec)(caγ) ≤c(ec(ea)(eγ)), where cr(c) = γ; to see this, note thatc(ec)(caγ) = ce(cc)(caγ)≤ce(cc)(ce(caγ))= ce(cc(caγ))< ce(c(caγ))by Lemma 6.4(a)= c(e(caγ))= c(ec(ea)(eγ)).

FINITE LEFT-DISTRIBUTIVE ALGEBRAS AND EMBEDDING ALGEBRAS25Clearly e(cr∗(a)) = cr∗(ea); since cr(aa) = a(cr(a)) > cr(a), this gives a(cr∗(a)) =cr∗(aa) > cr∗(a).Next, we define the new ternary relation ≡∗as follows: a ≡∗γ∗b iffa ≡δ b for someδ ∈O such that γ∗≤∗H(δ). In other words, a agrees with b up to some new ordinal iffa agrees with b up to some old ordinal at least as high.

Using this definition, it is easy todeduce all of the axioms about ≡∗from the corresponding axioms about ≡, except for theaxiom “if a ≡∗γ∗b and aδ∗< γ∗, then aδ∗= bδ∗”; this one will require more work.If cr(d) = δ, then H(δ) = [d, δ] ≤∗[a, δ] for any a, because dd(ddδ) = δ ≤d(daδ).Lemma 6.8. If cr(c) = γ, then [a, γ] ≤∗H(δ) if and only if caγ < cδ.Proof.

Fix d with critical point δ; then [a, γ] ≤∗[d, δ] is equivalent to cd(caγ) < c(ddδ) =cδ. It is clear that cd(caγ) < cδ implies caγ < cδ, because caγ ≤cd(caγ).

On the otherhand, if caγ < cδ, then caγ < cr(cd), so cd(caγ) = caγ < cδ.■Lemma 6.9. If a ≡γ b and [a, δ] ≤∗H(γ), then [a, δ] = [b, δ].Proof.

It is enough to show that [b, δ] ≤∗[a, δ], since then one can interchange a and b.Fix d such that cr(d) = δ. By the preceding lemma, we have daδ < dγ.

This allows us toconclude from da ≡dγ db that daδ = dbδ; since Lemma 6.4(a) gives dd(dbδ) < d(dbδ), weget dd(dbδ) < d(daδ), so [b, δ] ≤∗[a, δ], as desired.■We are now ready to prove the remaining property of ≡∗: if a ≡∗γ∗b and a[c, ρ] <∗γ∗,then a[c, ρ] = b[c, ρ]. Fix δ such that γ∗≤∗H(δ) and a ≡δ b.

We have [ac, aρ] <∗H(δ),so the statement preceding Lemma 6.8 gives H(aρ) < H(δ). Since H is order-preserving,we have aρ < δ.

Therefore, aρ = bρ, so, using ac ≡δ bc and Lemma 6.9, we get a[c, ρ] =[ac, aρ] = [bc, aρ] = [bc, bρ] = b[c, ρ].We have now completed the proof that E and O∗, together with the starred operationsand relations, form a two-sorted embedding algebra. Also, we have a canonical order-preserving map H from O to O∗, and it is easy to check that H sends all of the operationsand relations to their starred equivalents; hence, (E, O) is isomorphic to a subalgebra of(E, O∗), so (E, O∗) is isomorphic to an extension of (E, O).

It now remains to define theadditional operations of an extended two-sorted embedding algebra for (E, O∗).Since we want the pair [b, γ] to represent b(<γ), the formula a(

Fix c and c′ such that cr(c) = γ and cr(c′) = γ′. Since [b, γ] ≤∗[b′, γ′], we havecc′(cbγ) ≤c(c′b′γ′); applying c(c′a) to this gives c(c′a)(cc′(cbγ)) ≤c(c′a)(c(c′b′γ′)).

Butc(c′a)(cc′(cbγ)) = cc′(ca)(cc′(cbγ)) = cc′(ca(cbγ)) = cc′(c(a ◦b)γ)and c(c′a)(c(c′b′γ′)) = c(c′a(c′b′γ′)) = c(c′(a◦b′)γ′), so we have cc′(c(a◦b)γ) ≤c(c′(a◦b′)γ′)and hence [a◦b, γ] ≤∗[a◦b′, γ′].■So a(<[b, γ]) is well-defined. The next lemma shows that this definition matches theoriginal motivation.

26RANDALL DOUGHERTY AND THOMAS JECHLemma 6.11. For all a and γ, a(

Fix c such that cr(c) = γ; then a(

From γ < cγ, we get caγ < ca(cγ) = c(aγ), soLemma 6.8 gives [a, γ] ≤∗H(aγ). Therefore, Lemma 6.9 gives [a, γ] = [a◦c, γ], as desired.■We now verify that this definition of a(<γ∗) satisfies the first five axioms listed inDefinition 6.1.

Let γ∗= [c, ρ] and δ∗= [d, σ]. The first two axioms are proved by simplecomputations:a(b(<γ∗)) = a[b◦c, ρ] = [ab◦ac, aρ] = ab(<[ac, aρ]) = ab(

To provethis, fix r and s such that cr(r) = ρ and cr(s) = σ; thena(<γ∗) ≤∗aδ∗⇐⇒[a◦c, ρ] ≤∗[ad, aσ]⇐⇒r(as)(r(a ◦c)ρ) ≤r(as(ad)(aσ)⇐⇒ra(rs)(ra(rcρ)) ≤r(a(sdσ))⇐⇒ra(rs(rcρ)) ≤ra(r(sdσ))⇐⇒rs(rcρ) ≤r(sdσ)⇐⇒γ∗≤∗δ∗.For the fifth axiom, suppose a ≡∗γ∗b and a(<δ∗) ≤∗γ∗. Find η such that a ≡η band γ∗≤∗H(η); then a(<δ∗) ≤∗H(η) and a ◦d ≡η b ◦d, so Lemma 6.9 gives a(<δ∗) =[a◦d, σ] = [b◦d, σ] = b(<δ∗).It remains to find a suitable definition for the cofinality function.

Since [a, γ] is supposedto represent a(<γ), where γ is a critical point and hence regular, we define cf [a, γ] to beH(γ). As usual, we need a lemma showing that this does not depend on the choice of arepresentative for the equivalence class [a, γ].Lemma 6.12.

If γ ̸= δ, then [a, γ] ̸= [b, δ].Proof. We may assume γ < δ.

Fix c and d such that cr(c) = γ and cr(d) = δ; then cr(dc) =dγ = γ and cr(dcd) = dcδ ≥δ > γ. We can use dc instead of c when comparing [a, γ] with[b, δ]: [a, γ] ≤∗[b, δ] iffdcd(dcaγ) < dc(dbδ).

Now the assumption that [a, γ] = [b, δ] leads

FINITE LEFT-DISTRIBUTIVE ALGEBRAS AND EMBEDDING ALGEBRAS27to a contradiction as follows:dcd(dcaγ) < dc(dbδ)since [a, γ] ≤∗[b, δ]< d(caγ)since [b, δ] ≤∗[a, γ]= d(ca)(dγ)= dc(da)γ= dcd(dca)(dcdγ)= dcd(dcaγ).■It is now trivial to verify the axioms cf(cr∗(a)) = cr∗(a), cf(a(<γ∗)) = cf γ∗, andcf(aγ∗) = a(cf γ∗).The last two axioms can actually be deduced from the other axioms when the ordinal γis of the form b(<δ) where δ is a critical point; since every element of O∗has this form, thiswill suffice here. The law cf γ ≤γ follows from Proposition 6.2(1), since cf γ = cf δ = δ.Now suppose a does not move δ = cf γ; then a(

One example is the following result,which Laver proved for elementary embeddings (Theorem 2.14).In a two-sorted embedding algebra, let j ̸= id be some embedding, and let Aj be theset of embeddings generated from j by the operation · (so each a ∈Aj is given by a wordin WA).Theorem 6.13. Assume that the set of all critical points of elements of Aj has ordertype ω.

If a and b are distinct elements of Aj, then there is a critical point γ such thata(γ) ̸= b(γ).Proof. We may assume that all ordinals in the algebra are critical points; otherwise, justmove to the subalgebra comprising all embeddings and all critical points.

Apply Theo-rem 6.3 to construct an extended two-sorted embedding algebra which is an extension ofthe given algebra. We now follow the proof of Theorem 13 from Laver [7]; every step ex-cept one in this proof uses only properties of the extended ordinals which are listed in 6.1and 6.2, and hence works in the same way here.

The one exception is the use of the factthat a certain increasing sequence of critical points is cofinal in the set of all critical pointsof Aj; we have made this fact an assumption of the theorem. The result is that, in theextended algebra, there exists a critical point γ such that aγ ̸= bγ.

But all critical pointsin the extended algebra are critical points in the original algebra (since the same holds forembeddings), so so we have the desired result in the original algebra.■

28RANDALL DOUGHERTY AND THOMAS JECH7. Construction of an embedding algebraIn this section, we will prove one direction of Theorem 5.2 by showing how to con-struct an embedding algebra under the assumption that A∞is free (and hence all of thestatements in Theorem 4.4 hold).We will first construct a two-sorted embedding algebra.

The embedding set E will beP∞∪{0}, while the ordinal set O will be ω. The operations · and ◦on E will of course bethose obtained from P∞, and 0 will be the identity in E.We note that 4.4(vi) implies the stronger statement that, for every a ∈WP, there isan n such that [a]n ̸= 0.

To see this, use Lemma 2.11 to find a word in WP of the forma1 ◦· · · ◦ak (a1, . .

., ak ∈WA) which is equivalent to a. By 4.4(vi), there exists n so largethat [ai]n ̸= 0 for all i; then formula (3.3) implies that [a]n ̸= 0.For each a ∈WP, define the function ea: ω →ω as follows: for each n ∈ω, letea(n) = s(a · 2n).

In other words, ea(n) is the largest m such that [a · 2n]m = 0. (By thestrengthened 4.4(vi), there is a largest such m for each n.) If a = b in P∞, then [a]m = [b]mand [a · 2n]m = [a]m ∗[2n]m = [b]m ∗[2n]m = [b · 2n]m for all n and m; hence, ea = eb.It therefore makes sense to write ea for a ∈P∞.

This will give the desired applicationfunction from E × O to O, so we will sometimes write a(n) for ea(n) (but not an, as thismight be confused with a · n). Define e0 to be the identity function on ω.For any a ∈P∞, we can apply Proposition 3.2 to show that, if [a · 2n]m = 0, then[a · 2n+1]m+1 = 0; it follows that the function ea is strictly increasing.

(This is obviouslytrue for e0 as well.) Now induction gives ea(n) ≥n for all n.Next, we prove that ea◦b = ea ◦eb (i.e., the algebra operation ◦represents composition).This follows from the corollary to Lemma 4.3:ea◦b(n) = s((a ◦b) · 2n) = s(a · (b · 2n)) = s(a · 2s(b·2n)) = s(a · 2eb(n)) = ea(eb(n)).For a ∈P∞, define cr(a) to be the largest m such that [a]m = 0, as given by thestrengthened 4.4(vi).

(We will see later that this is the critical point of ea.) It follows that[a]m+1 = 2m, so [a·2m]m+1 = [2m·2m]m+1 = 0.

(For the last equality, see Proposition 3.2. )This proves that a(cr(a)) > cr(a).We now define ≡N for N ∈O by: a ≡N b iff[a]N = [b]N. The fact that the algebra PNsatisfies (LL) immediately implies most of the desired properties of ≡N.

In particular, ifa ≡N b and a(m) < N, then [a·2m]N is nonzero, and [b·2m]N must have the same nonzerovalue, so we find that a(m) = b(m). The only remaining property that is nontrivial iscoherence, for which we argue as follows.

Suppose [a]N = [b]N and M = ec(N); we mustshow that [ca]M = [cb]M. The definition of M implies that [c·2N]M = 0, so the period of cin PM divides 2N. But [a]N = [b]N, so [a]M and [b]M are congruent modulo 2N; therefore,[ca]M = [cb]M, as desired.This completes the construction of the two-sorted embedding algebra.

The point of con-structing this intermediate algebra is that it allows us to apply Theorem 6.13 to concludethat, if a ̸= b in A∞, then ea ̸= eb.We now construct an embedding algebra as follows. Let A = {ea: a ∈A∞}.

Define theoperation · on A by the formula ea · eb = eab; this definition is valid because the mapping

FINITE LEFT-DISTRIBUTIVE ALGEBRAS AND EMBEDDING ALGEBRAS29from a to ea is one-to-one. It is clear that A is generated from the single function e1 bythe operation ·.Proposition 5.4(1) implies that the critical point of ea is equal to the number cr(a)defined above.

Given this, it is easy to see that A satisfies the axioms of an an embeddingalgebra by using the corresponding properties of the two-sorted embedding algebra. Thiscompletes the construction.8.

Uniqueness of embedding algebrasIn this section, we will prove the following uniqueness result for monogenic embeddingalgebras.Theorem 8.1. (a) If (A, ·) is a monogenic embedding algebra for which every naturalnumber is a critical point, then (A, ·) is isomorphic to the embedding algebra constructedfrom P∞in the preceding section.

(b) If (E, O; ·, ◦, . .

.) is a monogenic two-sorted embedding algebra in which the ordinalshave order type ω and every ordinal is a critical point, then it is isomorphic to the two-sorted embedding algebra constructed from P∞in the preceding section.Along the way, we will show that, if a nontrivial embedding algebra (or a nontrivialtwo-sorted embedding algebra with ordinals of order type ω) exists, then 4.4(vi) holds,and hence A∞is free, thus completing the proof of Theorem 5.2.

Most of the argumentsin this section are adapted from Laver [7].If an embedding algebra satisfies the hypotheses of Theorem 8.1(a), then, as noted afterthe proof of Proposition 5.5, we can expand/extend it to a two-sorted embedding algebraas hypothesized in Theorem 8.1(b). So let us assume we have such a two-sorted embeddingalgebra.

Let j be the generating embedding, and let Aj be the set of embeddings generatedfrom j using · alone. As noted in section 5, the set of non-identity embeddings is closedunder ·, so every element of Aj has a critical point.

For any a ∈WA, let ja be the result ofreplacing each 1 in the expression a with j. (Note that Aj = {ja: a ∈WA}.) In particular,since we identified positive integers with words in WA, we have an embedding jm for eachm > 0, and j1 = j; also, we let j0 = id.Let γn be the critical point of j2n.

Recall that, for any a ∈WA, [a]n is defined to bethe result of evaluating a in An = {0, 1, . .

., 2n −1}.Proposition 8.2. For any a ∈W ∗A, ja ≡γn j[a]n.Proof.

Since j2n ≡γn id = j0, it does not matter whether we work with An or A′n ={1, . .

., 2n}. Clearly the proposition holds for a = 0.

We will show that, for any b, c ∈A′n,jbjc ≡γn jb∗nc; given this, an easy induction on a ∈WA yields the proposition.The proof of jbjc ≡γn jb∗nc is by double induction, downward on b and upward on c.For b = 2n, we have j2njc ≡γn id · jc = jc = j2n∗nc. The case b < 2n, c = 1 is also trivial:jbj1 = jb+1 = jb∗n1.

Finally, for b, c < 2n,jbjc+1 = jb(jcj) = (jbjc)(jbj) ≡γn jb∗ncjb+1 ≡γn j(b∗nc)∗n(b+1) = jb∗n(c+1).

30RANDALL DOUGHERTY AND THOMAS JECH(The induction hypothesis can be used in the second-to-last step because b ∗n c > b.) Thiscompletes the induction.■Proposition 8.3.

For all n, γn < γn+1; also, for all m > 0, cr(jm) = γk where 2k is thelargest power of 2 dividing m.Proof. By induction on N, we show that these statements are true for n < N and m < 2N.The case N = 0 is vacuous.Suppose now that the assertion is true for N; we willprove it for N + 1.

We know that γ0 < γ1 < · · · < γN. By definition, cr(jm) = γNif m = 2N.

If 2N < m < 2N+1, then Proposition 8.2 implies that jm ≡γN jm−2N, so,if 2k is the largest power of 2 dividing m −2N, then 2k is also the largest power of 2dividing m, and cr(jm−2N) = γk < γN, so cr(jm) = γk. This means that the embeddingsjm for 2N < m < 2N+1 all have critical points below γN, and hence, by Proposition 5.7,jm(γN) > γN; let θ > γN be the least of these values jm(γN).Now coherence givesjm+1 = jmj ≡θ jm(j2N j) for all m in this range, soj2N+1 ≡θ j2N+1−1(j2N j)≡θ j2N+1−2(j2N j)(j2Nj)≡θ .

. .≡θ j2N+1(j2Nj) · · ·(j2N j) = j2N j2N .Since cr(j2Nj2N ) = j2N(γN) > γN and γN < θ, we must have cr(j2N+1) > γN.Thiscompletes the induction.■We can now show that 4.4(vi) holds, and hence A∞is free, as follows: Suppose a ∈WA.Since the sequence of critical points γn is strictly increasing, and the ordinals have ordertype ω, there must be an n such that cr(ja) < γn.

Then ja ̸≡γn id = j0, so we must have[a]n ̸= 0. This completes the proof of Theorem 5.2.Proposition 8.3 implies that jm ̸≡γn id for 1 ≤m < 2n (because cr(jm) < γn).

Con-sequently, we have jm ̸≡γn jm′ for 1 ≤m < m′ ≤2n; if this were not so, then one couldapply jm and jm′ to j 2n −m′ times to get jm+2n−m′ ≡γn j2n ≡γn id, a contradiction.It follows that the mapping ja/≡γn 7→[a]n from Aj/≡γn to An is bijective and preservesthe operation ·, so it is an isomorphism. These mappings commute with the canonicalprojections from Aj/≡γn+1 to Aj/≡γn and from An+1 to An, so they give a mapping fromAj to the inverse limit of the algebras An; clearly this mapping sends the generator of Ajto the generator of A∞, so we have a mapping f from Aj onto A∞.

Since f preserves ·,and since A∞is free, f must be an isomorphism between Aj and A∞.For any a ∈WA, if n is so large that cr(ja) < γn, then Proposition 8.2 gives ja ≡γn j[a]n,and [a]n must be nonzero, so, by Proposition 8.3, cr(ja) = γk where 2k is the largest powerof 2 dividing [a]n. This k is just s(a). Also, for any m, we getja(γm) = ja(cr(j2m)) = cr(jaj2m) = cr(ja·2m) = γs(a·2m) = γea(m),

FINITE LEFT-DISTRIBUTIVE ALGEBRAS AND EMBEDDING ALGEBRAS31where ea is as defined in section 7. Finally, for any a, b ∈W ∗A, we haveja ≡γn jb ⇐⇒j[a]n ≡γn j[b]n ⇐⇒[a]b = [b]n.Therefore, the structure of Aj is determined completely except for the possible existenceof ordinals which are not critical points.

(Even for these, the equivalence relation ≡δ isdetermined; the argument of the preceding paragraph shows that, if γn−1 < δ ≤γn, thenja ≡δ jb if and only if [a]n = [b]n.) In the situation of Theorem 8.1(a), there are no suchextra ordinals, and we have γn = n for all n; we can now see that the structure of Aj(which is just a copy of the original embedding algebra A) exactly matches the structuredefined in section 7 from P∞. So Theorem 8.1(a) is proved.Now, in the situation of Theorem 8.1(b), let Pj be the set of embeddings generatedfrom j using both · and ◦.

(Since the algebra is generated by j, this is all embeddingsexcept id.) In order to show that composition here matches the structure from section 7,we use the following result.Proposition 8.4.

If a, b ∈Aj and n ∈ω, then there is c ∈Aj such that a ◦b ≡γn c.Proof. We may assume that cr(a) > cr(b); otherwise, replace a and b with ab and a (usinga ◦b = ab ◦a).

Now let a0 = b, a1 = a, and ai = ai−1ai−2 for i ≥2. Induction givesai+1 ◦ai = a ◦b, cr(b) = cr(a0) = cr(a2) = cr(a4) = .

. .

, and cr(a) = cr(a1) < cr(a3)

..Since the sequence cr(a2i+1) is a strictly increasing sequence of criticalpoints, and the set of all critical points has order type ω, there must be an odd i such thatcr(ai) ≥γn; this gives a ◦b = ai ◦ai−1 ≡γn ai−1, so we can let c = ai−1.■It follows that, if a, b ∈An, then there is c ∈An such that ja ◦jb ≡γn jc; we know fromthe above results that this c is unique. To determine what c is, note that (ja ◦jb)j ≡γn jcj,so ja∗n(b+1) = jc+1, so a ∗n (b + 1) = c + 1, where the additions are performed modulo 2nin An; hence, c = (a ∗n (b + 1)) −1 = a ◦n b.

We therefore have ja ◦jb ≡γn ja◦nb fora, b ∈An; now, if we define ja for a ∈WP as we did for a ∈WA, then induction on agives ja ≡γn j[a]n for all a ∈WP. We can now argue as before that Pj/≡γn is isomorphicto Pn and Pj is isomorphic to P∞, so the structure of Pj is unique except for the possibleexistence of ordinals which are not critical points, and matches that from section 7.

Thiscompletes the proof of Theorem 8.1.One can in fact construct an embedding algebra with numbers that are not criticalpoints, either by just duplicating every critical point or, less trivially, by constructing theextended algebra in section 6 and then using the method of section 7 to convert this toan embedding algebra. (One can then modify the algebra further to get an embeddingalgebra which does not satisfy ab(a(n)) = a(b(n)).) For the less trivial construction, onemust observe that the ordinals in the extended algebra have order type ω.

To see this,note that if a ≡γn b and a(<κ) < γn, then a(<κ) = b(<κ); hence, there are at most n2nextended ordinals below γn.On the other hand, we now have a roundabout proof that, if there is a nontrivialembedding algebra, then there is one in which all natural numbers are critical points (and

32RANDALL DOUGHERTY AND THOMAS JECHhence ab(a(n)) = a(b(n)) holds), namely the one constructed from P∞. One would expectto be able to prove this directly, by simply deleting the natural numbers which are notcritical points and relabeling the critical points as 0, 1, 2, .

. ..

However, it is conceivablethat distinct functions in the algebra are the same when restricted to the critical points,so that · could fail to be well-defined after the other numbers are deleted. It turns out thatthis does not happen in the monogenic case, but the authors do not see a way to provethis without building up enough structure to imitate Laver’s proof of Theorem 2.14.9.

The strength of “A∞is free”As we recalled in section 1, Laver’s proof of the irreflexivity of the free left distribu-tive algebra on one generator assumed the existence of a nontrivial elementary embeddingfrom Vλ to itself; this is an extremely strong large cardinal hypothesis. (Actually, Laverhad noted that, since one only needs a bounded part of Vλ to talk about the finitely manyembeddings mentioned while comparing two given words in the free algebra, the assump-tion can be reduced to the existence of an n-huge cardinal for each natural number n.)The possibility that the irreflexivity property was strong enough to require large cardi-nal assumptions for its proof remained until Dehornoy proved the property without suchassumptions (in fact, using only Primitive Recursive Arithmetic).We now consider the statement “A∞is free” and the equivalent versions in Theorem 4.4.These statements imply that A∞is both free and irreflexive, so the irreflexivity of the freealgebra follows immediately.

The purpose of this section is to show that the statement“A∞is free” is strictly stronger than the statement “the free algebra is irreflexive,” in thefollowing sense:Theorem 9.1. The statement “A∞is free” is not provable in Primitive Recursive Arith-metic.Of course, we assume throughout that PRA is itself consistent.Proof.

It is a well-known result from proof theory (see Sieg [8]) that the only recursivefunctions that can be proved to be total using only PRA are the primitive recursive func-tions. Therefore, to prove the theorem, it will suffice to show that PRA + 4.4(vii) provesthe totality of a recursive function F which is not primitive recursive.For each natural number n, let F(n) be the largest m such that [un]m = 0, where unis the word 1 · (1 · (.

. .

(1 · 1) . .

.)) with n + 1 1’s.

It follows from 4.4(vii) that F is atotal recursive function. If the functions ea are defined as in section 7, thus giving anembedding algebra, then F(n) = en1 (0), so F is the critical sequence of the mapping e1.Since all natural numbers are critical points in this embedding algebra, one can state thatF(n) is the number of critical points below en1(0).We now use the methods of Dougherty [5] for producing many critical points.

Thatpaper is written in terms of elementary embeddings, but it is not hard to check that theonly properties used in section 2 of that paper are that each embedding gives a strictlyincreasing monotone function on the ordinals and that, if a and b are two such embeddings,then cr(ab) = a(cr(b)) and a(bγ) = ab(aγ) for all ordinals γ. Hence, the main theorem

FINITE LEFT-DISTRIBUTIVE ALGEBRAS AND EMBEDDING ALGEBRAS33of that paper, that the number of critical points below κn = jn(κ0) grows so rapidlywith n that it cannot be primitive recursive, applies to any nontrivial embedding algebraor two-sorted embedding algebra. (The results in later sections of that paper use onlythe properties of an extended two-sorted embedding algebra, so the stronger lower boundsobtained there also apply to any nontrivial embedding algebra or two-sorted embeddingalgebra.) But, in the embedding algebra from section 7, the number of critical pointsbelow κn is just F(n) as defined above, so F is not primitive recursive.■On the other hand, the freeness of A∞follows from the existence of a nontrivial el-ementary embedding j: Vλ →Vλ.

The proof of this (due to Laver) uses Theorem 2.13.Given this theorem, we can apply the arguments in section 8 to the monogenic two-sortedembedding algebra obtained from Pj to conclude that A∞is free. Laver (personal com-munication) has recently noted, and the authors have confirmed, that one can use themethod of proof of Theorem 2.13 while working with only an n-huge embedding, to get acorrespondingly weaker result; hence, the freeness of A∞follows from the existence of ann-huge cardinal for each natural number n. (There is a level-by-level form of this result:if a k-huge cardinal exists, then there is a natural number n such that [uk]n ̸= 0.

)The proof of Theorem 9.1 showed that the assumption that A∞is free can be used toconstruct a particular function F which grows too rapidly to be primitive recursive. Itturns out that one cannot produce any function growing much faster than F from thisassumption.

This can be stated precisely as follows.Proposition 9.2. Any recursive function which is provably total in PRA + “A∞is free”must grow more slowly than Fm for some m, where F0 = F and Fm+1 is the iterationof Fm (starting at 1, say; that is, Fm+1(n) = F nm(1)).Proof.

The proofs in Sieg [8] can be modified to give the following extended version of theproof-theoretic result used earlier:If P(n, m) is a primitive recursive predicate, f(n) is the least m such thatP(n, m) holds, and g is a recursive function which is provably total in PRA +∀n∃mP(n, m), then g can be obtained from f and trivial functions (constants,projections, and successor) by composition and primitive recursion.The function F can be used as f, since P(n, m) can be defined to be “un ̸= 0 in Am+1.”Also, 4.4(vii) is a consequence of PRA+∀n∃mP(n, m). Therefore, any recursive function gprovably total from PRA + “A∞is free” must be obtainable from F and trivial functionsby the operations of composition and primitive recursion.Now the standard proof byinduction on the number of such operations used shows that g is below Fn for some n.■As a particular case of this, recall that there is a primitive recursive algorithm forcomparing two expressions a and b, i.e., transforming them into equivalent expressions a′and b′ such that either a′ = b′ or one of a′, b′ is a left subterm of the other.

Starting withthis, one can go through Laver’s proof that any two distinct embeddings must differ ata critical point, and verify that all of the steps are primitive recursive. Hence, assumingA∞is free, if a and b are members of WA such that a ̸≡A b, and n is least such that ja

34RANDALL DOUGHERTY AND THOMAS JECHand jb differ at critical point number n, then n can be obtained from a and b by a functionwhose growth rate is comparable to that of F.10. Open problems and acknowledgmentsThere remain a number of open problems related to these algebras.

The main one, ofcourse, is the exact strength of the statement “A∞is free”; the gap between “more thanPRA” and “there is an n-huge cardinal for each n” is rather large.One can also askwhether “there is a nontrivial two-sorted embedding algebra” is as strong as “there is anontrivial embedding algebra.”It is still open whether Laver’s result on distinguishing elementary embeddings by theirbehavior on critical points (Theorem 2.14) can be extended to Pj. If it can, by methodsformalizable in an extended two-sorted embedding algebra, then one can define a versionof embedding algebra which includes a composition operation, and the existence of anontrivial such algebra will still be equivalent to “A∞is free.”Another area of interest is further extensions of the results in section 6 to includemore of the ordinals that can be defined from elementary embeddings.

(Eventually onemight hope to start with the embedding algebra obtained from A∞and construct a largerstructure including all of the important features of the algebra obtained from an elementaryembedding from Vλ to itself.) A natural next step is to try to define ordinals of the form“the least α such that a(α) ≥γ” for a given embedding a and ordinal γ.

Such ordinals seemto be closely tied to the inequality aa(γ) ≤a(γ): the existence of the ordinals allows oneto prove that the inequality holds, and the authors can show under the assumption of theinequality that there is a natural extension of a given monogenic two-sorted embeddingalgebra in which all ordinals are critical points to an algebra including such ordinals.The authors do not yet have a large-cardinal-free proof that the inequality holds in theembedding algebra constructed from A∞, even assuming that A∞is free.The authors would like to thank P. Dehornoy for discussing his work and suggestingfurther questions, R. Laver for showing us his unpublished results, M. Rathjen for con-sultations about proof theory, and J. Zapletal for pointing out the construction used inProposition 2.12.References1. P. Dehornoy, Free distributive groupoids, J.

Pure Appl. Algebra 61 (1989), 123–146.2., Sur la structure des gerbes libres, C. R. Acad.

Sci. Paris S´er.

I Math. 309 (1989), 143–148.3., The adjoint representation of left distributive structures, Comm.

in Algebra 20 (1992), 1201–1215.4., Braid groups and left distributive operations, Trans. Amer.

Math. Soc.

345 (1994), 115–150.5. R. Dougherty, Critical points in an algebra of elementary embeddings, Ann.

Pure Appl. Logic 65(1993), 211–241.6.

R. Laver, The left distributive law and the freeness of an algebra of elementary embeddings, Adv.Math. 91 (1992), 209–231.7., On the algebra of elementary embeddings of a rank into itself, Adv.

Math. 110 (1995), 334–346.8.

W. Sieg, Fragments of arithmetic, Ann. Pure Appl.

Logic 28 (1985), 33–71.

FINITE LEFT-DISTRIBUTIVE ALGEBRAS AND EMBEDDING ALGEBRAS359. F. Wehrung, Gerbes primitives, C. R. Acad.

Sci. Paris S´er.

I Math. 313 (1991), 357–362.Department of Mathematics, Ohio State University, Columbus, OH 43210E-mail address: rld@math.ohio-state.eduPennsylvania State University, 215 McAllister Building, University Park, PA 16802E-mail address: jech@math.psu.edu

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