The abridged version of the paper containing all the research on generalizations of Laver tables which I have done on and off for about two years is now posted here. This version contains all definitions, theorems, but I have omitted the proofs of the theorems since I am still editing the proofs. The unabridged version of the paper should be about 135 pages or so.

From the mid 1990’s to about 2012, no results have been published on Laver tables or the quotient algebras of elementary embeddings. Nevertheless, set theorists have considered the algebras of elementary embeddings to be important enough that they have devoted Chapter 11 in the 24 chapter Handbook of Set Theory to the algebras of elementary embeddings.

Since I am the only one researching generalizations of Laver tables, and only 3 other people have published on Laver tables since the 1990’s, I have attempted to make the paper self-contained and readable to a general mathematician.

Any comments or criticism either by e-mail or on this post about the paper would be appreciated.

Let me now summarize some of the results from the paper.

- The double inductive definition of a Laver table extends greatly to a universal algebraic context where we may have some of the following:
- The algebra may be generated by many elements (multigenic Laver tables).
- The algebra may have many fundamental operations (multi-Laver tables).
- The algebra may have fundamental operations of higher arity (endomorphic Laver tables).
- Only some of the fundamental operations are self-distributive (partially endomorphic Laver tables).
- The fundamental operations satisfy a twisted version of self-distributivity such as

$$t(a,b,c,t(w,x,y,z))$$ $$=t(t(a,b,c,w),t(b,c,a,x),t(c,a,b,y),t(a,b,c,z))$$

(twistedly endomorphic Laver tables).

- The notion of a permutative LD-system algebraizes most of the properties that one obtains from the quotient algebras of elementary embeddings. For instance, permutative LD-systems have a notion of a composition operation, critical point, and equivalence up to an ordinal and these notions satisfy most of the properties that algebras of elementary embeddings satisfy. These notions are not contrived to look like the algebras of elementary embeddings since the critical point, composition, and equivalence up to an ordinal are fundamental to the theory of permutative LD-systems. Every quotient algebra of elementary embeddings $\mathcal{E}_{\lambda}/\equiv^{\gamma}$ is a permutative LD-system but the converse does not hold. The notion of a permutative LD-system provides a natural framework by which one can prove results about finite structures using large cardinal hypotheses.
- The set $(A^{\leq 2^{n}})^{+}$ consisting of all non-empty strings from the alphabet $A$ of length at most $2^{n}$ can be endowed with a unique self-distributive operation $*$ such that $\mathbf{x}*\mathbf{y}=\mathbf{y}$ whenever $|\mathbf{x}|=2^{n}$ and where $\mathbf{x}*a=\mathbf{x}a$ whenever $|\mathbf{x}|<2^{n}$ ($((A^{\leq 2^{n}})^{+},*)$ is a multigenic Laver table). The operation $*$ can be computed on a standard computer as long as $n$ is at most 26 or so (if $n=26$, then the output $\mathbf{x}*\mathbf{y}$ could have length up to $2^{26}$ so we run into some memory issues). All of the combinatorial information in the operation $*$ is contained within the classical Laver table $A_{n}$ along with a function $FM_{n}^{-}:\{1,...,2^{n}\}\times\{1,...,2^{n}\}\rightarrow\mathbb{Z}\setminus\{0\}$ called the final matrix. Entries in $FM_{n}^{-}$ can generally be computed on a standard computer in milliseconds as long as $n\leq 48$ (we reach the limit $n\leq 48$ since $A_{48}$ is the largest classical Laver table ever computed.). While the entries in $FM_{n}^{-}$ can be computed fairly quickly when one knows the classical Laver table $A_{n}$, the final matrices are combinatorially complex objects which contain intricacies which are not present in the classical Laver tables. The positive entries of the final matrix presumably form a subset of the Sierpinski triangle. There are many phenomena in the final matrices which I have observed experimentally but which have not been proven mathematically and which currently do not have any explanation.
- The multigenic Laver tables $(A^{\leq 2^{n}})^{+}$ form an inverse system. If for all $n$, there exists an $n$-huge cardinal, then almost every finite or countable subset of $\varprojlim_{n}(A^{\leq 2^{n}})^{+}$ freely generates a free LD-system (and even a free LD-monoid). Recall that Richard Laver has proven in the 1990’s that the free left-distributive algebra on
*one*generator can be embedded into an inverse limit of classical Laver tables. I have therefore constructed the machinery to prove an analogous result for free left-distributive algebras on multiple generators. - We present methods of producing new finite permutative LD-systems from old finite permutative LD-systems including
- extending a multigenic Laver table to a larger multigenic Laver table with one more critical point, and
- extending a permutative LD-system to a larger permutative LD-system by adding one point at a time.

- Not every finite reduced permutative LD-system arises as a subalgebra of a quotient of an algebra of elementary embeddings. This counterexample suggests that there may be an inconsistency in the large cardinal hierarchy first appearing somewhere between the huge cardinals and Berkeley cardinals (and hopefully at the level above I0).
- While the output of the endomorphic Laver table operations is typically very large. One can still compute information about the output of the endomorphic Laver table operations. As one would expect, the endomorphic Laver tables exhibit a sort of combinatorial complexity beyond the classical and multigenic Laver tables.
- Non-abelian group based cryptosystems such as the Anshel-Anshel-Goldfeld and Ko-Lee key exchanges extend to self-distributive algebras and also endomorphic algebras. The endomorphic Laver tables may be a good platform for such cryptosystems, but I do not know much about the security of the endomorphic Laver table based cryptosystem, and I have much uneasiness about the security of such cryptosystems even though I have not yet launched an attack (I have an idea of how an attack may work though).