Are there examples of algebraic structures which have been constructed using evolutionary algorithms and possibly machine learning algorithms? I am looking for algebraic structures like lattices rather than combinatorial structures that one finds in something like Ramsey theory.

Motivation

Evolutionary algorithms can be used in Ramsey theory to find lower bounds of Ramsey numbers by constructing graphs without the forbidden homogeneous subsets. Furthermore, I have used evolutionary algorithms to produce algebraic structures related to set theory.

In this question, I asked about how well one can make algebraic structures that satisfy certain identities using evolutionary algorithms that will always converge and never fall into local maxima. However, the condition that no local maxima exists is quite a strong condition, and evolutionary algorithms may still be useful even if it is possible or probable to fall into a local maximum. Furthermore, there may be good techniques that allow one to get out of the local maximum.

Details

Suppose that $E$ is a collection of structures and $P$ is a property of algebraic structures. Suppose that $s:E\rightarrow[0,1]$ is an easily computable function such that $\mathcal{X}$ satisfies $P$ if and only if $s(\mathcal{X})=1$ and where $s(\mathcal{X})$ is a measure of how close $\mathcal{X}$ is to satisfying $P$. Then the following algorithm template may be used to produce a sequence of structures $(\mathcal{X}_{0},\dots,\mathcal{X}_{N},\dots)$ that terminates with a structure that satisfies $P$.

Let $\mathcal{X}_{0}\in E$ be a structure. For each $n$, let $\mathcal{Y}_{n}$ be a slight and random modification of the structure $\mathcal{X}_{n}$. If $s(\mathcal{Y}_{n})\geq s(\mathcal{X}_{n})$, then set $\mathcal{X}_{n+1}=\mathcal{Y}_{n}$, otherwise set $\mathcal{X}_{n+1}=\mathcal{X}_{n}$. Once $\mathcal{X}_{n}\in P$, return $\mathcal{X}_{n}$.

Of course, one would need to modify the above algorithm template in case one falls into a local maximum.

Examples

We shall now look at two examples of how evolutionary algorithms can be used to produce algebraic structures.

Semigroups

Let $X$ be a set. Then the following algorithm can be used to produce associative operations on $X$:

Let $*_{0}$ be a random binary operation on $X$. For each $n$, let $\#_{n}$ be an operation on $X$ which is randomly selected according to the condition that $$|\{(x,y)|x\#_{n}y=x*_{n}y\}|=1.$$ If $$P((x\#_{n}y)\#_{n}z=P(x\#_{n}(y\#_{n}z))\geq P((x*_{n}y)*_{n}z=P(x*_{n}(y*_{n}z)),$$ then set $*_{n+1}=\#_{n}$ and otherwise set $*_{n+1}=*_{n}$. Once $*_{n}$ is associative, return $(X,*_{n})$ as the obtained associative structure.

Linear orderings on Laver tables

I have personally been able to produce linear orderings $\leq$ on the classical Laver tables $(A_{N},*)$ that satisfy the implication $y\leq z\rightarrow x*y\leq x*z$ using evolutionary algorithms. Such linear orderings on the classical Laver tables may be used to produce endomorphic Laver tables. Consider the following algorithm for producing linear orderings $\leq$ on $A_{N}$.

Let $\leq_{0}$ be a random linear ordering on $A_{N}$. For each $n$, let $\leq_{n}^{\sharp}$ be a randomly selected linear ordering on $A_{N}$ subject to the condition that there is a transposition $\pi\in S_{2^{n}}$ such that $x\leq_{n}y$ if and only if $\pi(x)\leq_{n}^{\sharp}\pi(y)$. If $$P(y\leq_{n}^{\sharp}z\Rightarrow x*y\leq_{n}^{\sharp}x*z)\geq P(y\leq_{n}z\Rightarrow x*y\leq_{n}x*z),$$ then set $\leq_{n+1}$ equal to $\leq_{n}^{\sharp}$ else set $\leq_{n+1}$ equal to $\leq_{n}$. Once the implication $y\leq_{n} z\rightarrow x*y\leq_{n} x*z$ is satisfied, return $\leq_{n}$.