Introduction to Laver tables and generalizations

The classical Laver tables are finite self-distributive algebras that arise from very large cardinals and thus establish a connection between the higher infinite and the finite. The classical Laver tables were initially discovered by Richard Laver in the late 1980’s in his investigation of rank-into-rank embeddings. I have been investigating several generalizations of the notion of a classical since July of 2015 including self-distributive algebras of arbitrary type.

Classical Laver tables and elementary embeddings

Suppose that $\lambda$ is a cardinal. Let $\mathcal{E}_{\lambda}$ denote the set of all elementary embeddings from $V_{\lambda}$ to $V_{\lambda}$. Define a binary operation $*$ on $\mathcal{E}_{\lambda}$ by letting $j*k=\bigcup_{\alpha<\lambda}j(k|_{V_{\alpha}})$. Then it is not difficult to show that $j*k\in\mathcal{E}_{\lambda}$ whenever $j,k\in\mathcal{E}_{\lambda}$. Furthermore, the algebra $\mathcal{E}_{\lambda}$ satisfies the self-distributivity identity $j*(k*l)=(j*k)*(j*l)$, and each non-trivial elementary embedding $j\in\mathcal{E}_{\lambda}$ freely generates a subalgebra of $\mathcal{E}_{\lambda}$ with respect to the self-distributivity identity.

Algebras which satisfy the identity $x*(y*z)=(x*y)*(x*z)$ are called LD-systems (these algebras have other names though), and these algebras also appear in the study of knots and braids and since LD-systems determine knots up to isomorphism and are thus useful in distinguishing knots.

If $\gamma<\lambda$ is a limit ordinal, then define an equivalence relation $\equiv^{\gamma}$ on $\mathcal{E}_{\lambda}$ by letting $j\equiv^{\gamma}k$ if and only if $j(x)\cap V_{\gamma}=k(x)\cap V_{\gamma}$ whenever $x\in V_{\gamma}$. Then $\equiv^{\gamma}$ is a congruence with respect to the operation $*$.

For each natural number $n$, there is a unique binary operation $*_{n}$ on $\{1,…,2^{n}\}$ such that $x*_{n}1=x+1$ whenever $x<2^{n}$, $2^{n}*_{n}1=1$, and $x*_{n}(y*_{n}z)=(x*_{n}y)*_{n}(x*_{n}z)$. The algebra $(\{1,…,2^{n}\},*_{n})$ is known as the $n$-th classical Laver table and it is denoted by $A_{n}$.

Theorem: Suppose that $j\in\mathcal{E}_{\lambda}$ is a non-trivial elementary embedding. Then the set of critical points $\{\text{crit}(k):k\in\langle j\rangle\}$ has order type $\omega$. If $\textrm{crit}_{n}(j)$ is the $n+1$-th element of $\{\text{crit}(k):k\in\langle j\rangle\}$, then $\langle j\rangle/\equiv^{\text{crit}_{n}(j)}\simeq A_{n}$.

Recall that a cardinal $\kappa$ is $n$-huge if there exists an elementary embedding $j:V\rightarrow M$ where $M$ is a transitive class, $crit(j)=\kappa$, and $M^{j^{n}(\kappa)}\subseteq M$ (i.e. $M$ is closed under sequences of length $j^{n}(\kappa)$). The $n$-huge cardinals are slightly below the rank-into-rank embeddings on the large cardinal heirarchy, but one can still form Laver tables from $n$-huge cardinals since $n$-huge embeddings are closed under taking sequences.

The following results are purely algebraic results about classical Laver tables whose only know proofs rely on large cardinal hypotheses.

Theorem: Suppose that for all $n$, there exists an $n$-huge cardinal. Then $(x*_{n}x)*_{n}y=2^{n}$ implies that $x*_{n}y=2^{n}$ as well.

If $n\in\omega$, then define a mapping $\phi_{n}:A_{n+1}\rightarrow A_{n}$ by letting $\phi_{n}(x)=x$ whenever $1\leq x\leq 2^{n}$ and $\phi_{n}(x)=x-2^{n}$ whenever $2^{n}<x\leq 2^{n+1}$. Then the system $(A_{n})_{n\in\omega}$ of classical Laver tables becomes an inverse system with transitional mappings $\phi_{n}$.

Theorem: Suppose that for all $n$, there exists an $n$-huge cardinal. The element $(1)_{n\in\omega}$ freely generates a sub-LD-system of the inverse limit $\varprojlim_{n\in\omega}A_{n}$.

The classical Laver tables are also significant for purely algebraic reasons since Ales Drapal has used the classical Laver tables to classify all LD-systems which are generated by a single element. Furthermore, it is likely that in the future, Laver tables will be used to produce deep results about knots and braids.

A need for a generalization of the classical Laver tables

Even though the classical Laver tables seem to be widely known and admired in the mathematical community, no progress was made towards understanding Laver tables from the beginning of the 21st century until the early 2010’s where couple papers on classical Laver tables have been published. Therefore, in order for the theory of Laver tables to sufficiently develop, it is necessary to introduce new structures and new ideas that are able to do the things that the classical Laver tables are unable to do by themselves.

While the free LD-systems on one generator can be embedded into the inverse limit of classical Laver tables, there does not seem to be a nice way to embed the free left-distributive algebra on even two generators into an inverse limit of classical Laver tables. The following result suggests that the free LD-system on two generators are more intricate than the free LD-system on one generator.

Theorem: The free LD-system on two generators is not isomorphic to a subalgebra of the free LD-system on one generator. However, the free LD-system on two generators has a free sub-LD-system on infinitely many generators.

Furthermore, while the classical Laver tables characterize subalgebras of $\mathcal{E}_{\lambda}/\equiv^{\gamma}$ generated by one element, the classical Laver tables do not characterize subalgebras of quotient algebras of elementary embeddings with multiple generators.  For example, if one assumes large cardinal hypotheses stronger than the existence of a non-trivial rank-into-rank embedding such as an $I1$-embedding in $\mathcal{E}_{\lambda}$, then $|\mathcal{E}_{\lambda}=2^{\lambda}$. Furthermore, the following result by Woodin gives examples in a forcing extension of rank-into-rank embeddings which generate disjoint algebras.

Theorem: (Woodin) Suppose that there is some rank-into-rank embedding. Then in some generic extension, there are $j,k\in\mathcal{E}_{\lambda}$ with $j\neq k$ but where $j|_{\lambda}=k|_{\lambda}$ and $\langle j\rangle\cap\langle k\rangle=\emptyset$.

Therefore, in order to study the algebraic structure of these algebras of elementary embeddings with many generators, one will need to use generalizations of Laver tables rather than simply the classical Laver tables.

Generalized Laver tables.

There are many classes of algebras which generalize the notion of a classical Laver table to other structures including generalized Laver tables, Laver-like LD-systems, locally Laver-like LD-systems, partially endomorphic Laver tables, Laver-like partially endomorphic algebras, etc.

Let $A$ be a set called an alphabet and let $A^{+}$ denote the set of all strings over the alphabet $A$. Let $\preceq$ be the prefix ordering on $A^{+}$ and let $L\subseteq A^{+}$ be a downwards closed subset. Let $M=\{\mathbf{x}a|\mathbf{x}\in L,a\in A\}\cup A$ and let $F=M\setminus L$. Suppose furthermore that $B^{+}\cap M$ is finite for each finite $B\subseteq A$. Then there exists a unique operation $*$ such that

  1. $\mathbf{x}*\mathbf{y}=\mathbf{y}$ for each $\mathbf{x}\in F$

2. $\mathbf{x}*a=\mathbf{x}a$ for each $\mathbf{x}\in L,a\in A$

3. $\mathbf{x}*(\mathbf{y}a)=(\mathbf{x}*\mathbf{y})*(\mathbf{x}a)$ for each $\mathbf{x}\in L,\mathbf{y}\in L$.

Then we shall call the algebra $(M,*)$ a pre-generalized Laver table. If $(M,*)$ is an LD-system, then we shall call $(M,*)$ a generalized Laver table. Most pre-generalized Laver tables are not self-distributive, but there are techniques for constructing new generalized Laver tables from existing generalized Laver tables.

Using generalized Laver tables, if one assumes strong large cardinal hypotheses, the free LD-systems on multiple generators can be approximated by finite LD-systems.

$\mathbf{Theorem}$ Assume that for all $n$ there exists an $n$-huge cardinal. Then the free LD-systems on two generators can be embedded into a product of generalized Laver tables.

Furthermore, almost every subalgebra of certain inverse limits of generalized Laver tables is free.

$\mathbf{Theorem}$ Assume that for all $n$, there exists an $n$-huge cardinal. Then there exists an inverse system $(M_{n})_{n\in\omega}$ of generalized Laver tables with the following property:

Let $I$ be a finite or countable set. Let $U=$

$$\{(\mathbf{x}_{i})_{i\in I}\in(\varprojlim_{n\in\omega}X_{n})^{I}|\text{$(\mathbf{x}_{i})_{i\in I}$ freely generates a subalgebra of $\varprojlim_{n\in\omega}X_{n}$}\}.$$

Then $U$ is a dense $G_{\delta}$-subset of $\varprojlim_{n\in\omega}X_{n}$.

With generalized Laver tables, one can easily deduce the following result about algebras of elementary embeddings.

Theorem: Suppose that $A$ is a finite set, $\lambda$ is a cardinal and $j_{a}\in\mathcal{E}_{\lambda}$ for each $a\in A$. Then whenever $\gamma<\lambda$, the subalgebra $\langle j_{a}\rangle/\equiv^{\gamma}$ of $\mathcal{E}_{\lambda}/\equiv^{\gamma}$ is finite. In particular, the set of critical points $\{\textrm{crit}(k)|k\in\langle j_{a}\rangle/\equiv^{\gamma}\}$ has order type $\omega$.

The generalized Laver tables very closely mimic the algebras of elementary embeddings; one can define the notion of a critical point on generalized Laver tables, and the reduced generalized Laver tables can be given an associative operation that resembles the composition of elementary embeddings. Furthermore, the notion of a critical point is quite essential to the theory of generalized Laver tables. In a generalized Laver table, one can also define the equivalence relation $\equiv^{\alpha}$ purely algebraically.

While the generalized Laver tables at first glance seem to characterize the kinds of algebras which one obtains from algebras of elementary embeddings, not all generalized Laver tables satisfy the correct properties in which algebras of elementary embeddings are known to satisfy. There exists a $64$ element generalized Laver table $M$ which I have obtained from computer calculations such that if $\simeq$ is a congruence on $M$ with $x\simeq y\rightarrow\textrm{crit}(x)=\textrm{crit}(y)$, then $M/\simeq$ is not isomorphic to any subalgebra of an algebra of the form $\mathcal{E}_{\lambda}/\equiv^{\gamma}$.

Endomorphic Laver tables.

I have generalized the notion of a Laver table to a universal algebraic setting so that one can obtain ternary and more generally Laver tables of an arbitrary type called ternary Laver tables, endomorphic Laver tables, and even partially endomorphic Laver tables. Like the generalized Laver tables, there are many examples of endomorphic Laver tables and partially endomorphic Laver tables. Unlike the generalized Laver tables, the partially endomorphic Laver tables are in general not locally finite.

A ternary operation $t$ on a set $X$ is said to be self-distributive if $t(a,b,t(x,y,z))=t(t(a,b,x),t(a,b,y),t(a,b,z))$ whenever $a,b,x,y,z\in X$. Clearly, a ternary operation $t$ is self-distributive if and only if the function $L_{a,b}:X\rightarrow X$ defined by $L_{a,b}(x)=t(a,b,x)$ is an endomorphism on $(X,t)$ whenever $a,b\in X$. Therefore, the ternary self-distributive operations produce many inner endomorphisms of the algebra. If $(X,t)$ is a ternary self-distributive algebra, then one can define a binary self-distributive operation $*$ on $X^{2}$ by $(a,b)*(x,y)=(t(a,b,x),t(a,b,y))$, and the algebra $(X^{2},*)$ is called the hull of $(X,t)$. Therefore, since the hull of a ternary self-distributive algebra is an LD-system, the ternary self-distributive algebras are a useful way to produce binary self-distributive algebras. The ternary Laver tables generalize the notion of a Laver table to ternary self-distributive algebras. The endomorphic algebras generalize the notion of self-distributivity to algebras of arbitary type, and the endomorphic Laver tables generalize the notion of a Laver table to endomorphic algebras of arbitary type. The partially endomorphic Laver tables are similar to the endomorphic Laver tables, but with partially endomorphic Laver tables, only some of the fundamental operations distribute over all of the terms.

Not much is known about the partially endomorphic Laver tables besides the definition of a partially endomorphic Laver tables and a general method of producing examples of partially endomorphic Laver tables. For example, it is not known whether the free ternary self-distribute algebras can be embedded into inverse limits of ternary Laver tables. Even though the endomorphic Laver tables are recursive whenever their underlying sets are recursive, the recursive definition of the endomorphic Laver tables is extremely slow on computers. My computer has much trouble computing even in an infinite ternary Laver table that corresponds to $A_{4}$. It seems like the length of the output of even the ternary Laver table function in some instances grows as quickly as the Ackermann function.

A philosophy behind Laver tables

The algebras of elementary embeddings are the only algebraic structures in set theory which are of a considerable interest to the broader mathematical community due to their finitistic algebraic nature. The Laver tables are the only well-known structures in modern set theory which are computable and which have been investigated through computer calculations, and the Laver tables arise from the highest levels of the large cardinal hierarchy. At this point in time, the study of the algebras of rank-into-rank embeddings embeddings seems to be one of the few parts of research level set theory which is still accessible to open-minded non set-theorists (although I doubt that this will remain the case in the future as the theory of algebras of elementary embeddings develops further). Furthermore, the study of the algebras of elementary embeddings is largely pure algebra. Once one establishes the Laver-Steel theorem and some basic facts about elementary embeddings, most of the main results about the algebras of elementary embeddings follow from purely algebraic theorems that do not make any reference to large cardinals.  The relation between large cardinals and Laver tables seems to even extend to the endomorphic Laver tables since quotient algebras of elementary embeddings can easily be made into partially endomorphic algebras which are used to produce partially endomorphic Laver tables.

It is too early to tell how the theory of Laver tables may develop in the near future because before 2015, the notion of a generalized Laver table has not been conceived. However, it seems like in the near future, the Laver tables will continue to provide a much needed strong connection between set theory and other areas of mathematics including algebra, knot theory, braid theory, universal algebra, and number theory. It seems like algebras of elementary embeddings will provide many results in these areas in which the large cardinal hypotheses cannot be removed. If the Laver tables are applicable to these areas of mathematics, then hopefully, the algebras of elementary embeddings will help instill an appreciation for set theory (in particular large cardinals) in non-set theorists and provide an avenue for collaboration between set-theory and different fields of mathematics.

If a mathematician were searching for an inconsistency in the large cardinal heirarchy, then the Laver tables would be the best place to look for such an inconsistency for a several reasons. First of all, the $n$-huge cardinals and rank-into-rank cardinals are near the top of the large cardinal hierarchy so they have a very high consistency strength which goes far beyond the realm of inner model theory, the Vopenka principle, and other large cardinal axioms whose consistency seems to have some reasonable justification. Furthermore, the rank-into-rank cardinals are just couple steps below the Kunen inconsistency, and there does not seem to be a known good philosophical reason why rank-into-rank cardinals should be consistent that does not also apply for the Kunen inconsistency as well. Also, it seems like the most likely scenario for an inconsistency to turn up in the large cardinal hierarchy is if large cardinals imply a finite or computable combinatorial object has a property that can be shown to be false, and for the time being, the only such finite combinatorial objects are the Laver tables. Finally, the fact that not all generalized Laver tables come from rank-into-rank embeddings hints at the possibility of an inconsistency in the upper reaches of the large cardinal heirarchy because such a counterexample indicates that the correspondence between rank-into-rank embeddings and algebraic structures is not as orderly and simple as one might hope it to be. That being said, I do not hope or expect there to be an inconsistency in the large cardinal hierarchy even at the level of rank-into-rank cardinals. If an inconsistency were to turn up in the large cardinal hierarchy, then I would be saddened by such a result. However, if the Laver tables predict many empirically falsifiable results, but no inconsistency is found in the upper reaches of the large cardinal hierarchy, then one would have strong reasons to accept these large cardinals as consistent and perhaps even true in some reasonable transitive model of set theory or true in the real universe of set theory.

I should also mention that the possibility of an inconsistency is not a good reason for one to not study rank-into-rank cardinals. Instead, the possibility of an inconsistency gives several reasons why people should investigate rank-into-rank cardinals. One is only able to make an educated guess about the consistency of rank-into-rank cardinals if rank-into-rank cardinals are thoroughly investigated. Furthermore, the fact that not even the consistency of large cardinals is provable in ZFC means that large cardinals go beyond ZFC and make predictions about the world that cannot be made otherwise. In other words, large cardinals provide an understanding to the universe that cannot be obtained by any other means. Another reason to study rank-into-rank cardinals is that finding an inconsistency around the level of rank-into-rank cardinals is an interesting mathematical problem in its own right which should be investigated (of course, here this problem is only a mathematics problem in the loose sense of the word since by the second incompleteness theorem one can only solve this problem if there is an inconsistency).

I am currently the only researcher studying generalizations of Laver tables but I expect for other researchers to investigate these generalizations of Laver tables because there are plenty of problems to work on regarding the generalizations of Laver tables.