We can use artificial intelligence to generate some bi-right-distributive idempotent left quasigroups $(X,*,+)$.

To do this, we first of all need to represent the algebraic structure $(X,*,+)$ in a way that is easy for the artificial intelligence to work with.

We shall represent $(X,*,+)$ as a system $(g_{x})_{x\in X}$ where $g_{x}:X\rightarrow X$ is the permutation for each $x\in X$ such that $x*y=g_{x}(y),x+y=g_{x}^{-1}(y)$ whenever $x,y\in X$. Here, $(X,*,+)$ satisfies the identities $x+(x*y)=y=x*(x+y)$ for any system of permutations $(g_{x})_{x\in X}$.

Now we define a loss function $L$ where the domain of $L$ is the set of all tuples
$(g_{x})_{x\in X}$ such that $g_{x}$ is a bijection from $X$ to $X$ for each $x\in X$ as follows:

$$L((g_{x})_{x\in X})=|\{x\in X\mid g_{x}(x)\neq x\}|$$
$$+\{(x,y,z)\mid(x*y)+z\neq(x+z)*(y+z)\}+\{(x,y,z)\mid(x+y)+z\neq(x+z)+(y+z)\}$$
$$+\{(x,y,z)\mid(x+y)*z\neq(x*z)+(y*z)\}+\{(x,y,z)\mid (x*y)*z\neq(x*z)*(y*z)\}.$$

The motivation for the loss function $L$ is that $L((g_{x})_{x\in X})$ is a measure of how close the algebra $(X,+,*)$ is to being a bi-right-distributive idempotent left quasigroups, and $L((g_{x})_{x\in X})=0$ if and only if $(X,+,*)$ is a bi-right-distributive idempotent left quasigroup.

Now, one can construct bi-right-distributive idempotent left quasigroups by starting off with a random system of permutations $(g_{x})_{x\in X}$ and then repeatedly modifying this system of permutations in such a way that $L((g_{x})_{x\in X})$ gets lower and lower. Eventually you will get $L((g_{x})_{x\in X})=0$.

Here is the code that produces such an algebra in the language GAP (this code is not optimized for speed but is instead designed to be simple; occasionally the algorithm gets stuck in a local minimum).

```
``````
loss:=function(l)
local c,i,j,k,f,g;
f:=function(x,y) return y^l[x]; end;
g:=function(x,y) return y^(l[x]^(-1)); end;
c:=0;
for i in [1..n] do if not f(i,i)=i then c:=c+1; fi;
for j in [1..n] do
for k in [1..n] do
if not f(f(i,j),k)=f(f(i,k),f(j,k)) then c:=c+1; fi;
if not g(g(i,j),k)=g(g(i,k),g(j,k)) then c:=c+1; fi;
if not g(f(i,j),k)=f(g(i,k),g(j,k)) then c:=c+1; fi;
if not f(g(i,j),k)=g(f(i,k),f(j,k)) then c:=c+1; fi;
od;
od;
od;
return c; end;
n:=10; l:=[]; for i in [1..n] do l[i]:=Random(SymmetricGroup(n)); od;
p:=loss(l);
while true do ll:=StructuralCopy(l); a:=Random([1..n]); r:=1; s:=1;
while r=s do r:=Random([1..n]); s:=Random([1..n]); od;
if Random([1..30])=1 then ll[a]:=ll[a]*(r,s);
else ll[a]:=(r,s)*ll[a]*(r,s); fi;
q:=loss(ll);
if q<=p then p:=q; l:=ll; fi;
Display(p);
if p=0 then break; fi;
od;
l;
```

You can also minimize the loss using the following evolutionary algorithm.

```
list:=[];
for i in [1..10] do list[i]:=[];
for j in [1..n] do list[i][j]:=Random(SymmetricGroup(n)); od;
od;
while true do
for i in [11..100] do
if Random([true,false]) then
qq:=StructuralCopy(list[Random([1..10])]);
else qq:=[]; for j in [1..10] do qq[j]:=list[Random([1..10])][j]; od;
fi;
a:=Random([1..n]);b:=1;c:=1;
while b=c do b:=Random([1..10]); c:=Random([1..10]); od;
if Random([true,false]) then qq[a]:=qq[a]*(b,c); else qq[a]:=(b,c)*qq[a]*(b,c); fi;
list[i]:=qq;
od;
SortBy(list,loss);
while Length(list)>10 do Remove(list); od;
qar:=List(list,loss); Display(qar);
if Sum(qar)=0 then break; fi;
od; list;
```

**An algebra produced using artificial intelligence**

Here is a right-distributive idempotent left quasigroup produced using our algorithms.

$\begin{array}{c|c|c|c|c|c|c|c|c|c|c|}
* &\mathbf{1}&\mathbf{2}&\mathbf{3}&\mathbf{4}& \mathbf{5}&\mathbf{6}&\mathbf{7}&\mathbf{8}&\mathbf{9}&\mathbf{10}\\
\hline
\mathbf{1}& 1& 9& 3& 4& 10& 6& 7& 8& 2& 5 \\
\hline
\mathbf{1}& 1& 2& 3& 8& 10& 7& 6& 4& 9& 5 \\
\hline
\mathbf{1}& 1& 2& 3& 8& 10& 7& 6& 4& 9& 5 \\
\hline
\mathbf{1}& 1& 9& 3& 4& 10& 6& 7& 8& 2& 5 \\
\hline
\mathbf{1}& 8& 9& 2& 1& 5& 6& 7& 4& 3& 10 \\
\hline
\mathbf{1}& 4& 2& 3& 8& 10& 6& 7& 1& 9& 5 \\
\hline
\mathbf{1}& 4& 2& 3& 8& 10& 6& 7& 1& 9& 5 \\
\hline
\mathbf{1}& 1& 9& 3& 4& 10& 6& 7& 8& 2& 5 \\
\hline
\mathbf{1}& 1& 2& 3& 8& 10& 7& 6& 4& 9& 5 \\
\hline
\mathbf{1}& 8& 9& 2& 1& 5& 6& 7& 4& 3& 10
\end{array}$

$\begin{array}{c|c|c|c|c|c|c|c|c|c|c|}
+ &\mathbf{1}&\mathbf{2}&\mathbf{3}&\mathbf{4}& \mathbf{5}&\mathbf{6}&\mathbf{7}&\mathbf{8}&\mathbf{9}&\mathbf{10}\\
\hline
\mathbf{1}& 1& 9& 3& 4& 10& 6& 7& 8& 2& 5 \\
\hline
\mathbf{2}& 1& 2& 3& 8& 10& 7& 6& 4& 9& 5 \\
\hline
\mathbf{3}& 1& 2& 3& 8& 10& 7& 6& 4& 9& 5 \\
\hline
\mathbf{4}& 1& 9& 3& 4& 10& 6& 7& 8& 2& 5 \\
\hline
\mathbf{5}& 4& 3& 9& 8& 5& 6& 7& 1& 2& 10 \\
\hline
\mathbf{6}& 8& 2& 3& 1& 10& 6& 7& 4& 9& 5 \\
\hline
\mathbf{7}& 8& 2& 3& 1& 10& 6& 7& 4& 9& 5 \\
\hline
\mathbf{8}& 1& 9& 3& 4& 10& 6& 7& 8& 2& 5 \\
\hline
\mathbf{9}& 1& 2& 3& 8& 10& 7& 6& 4& 9& 5 \\
\hline
\mathbf{10}& 4& 3& 9& 8& 5& 6& 7& 1& 2& 10 \\
\hline
\end{array}$

**Generalization**

Since this question was motivated by emergent algebras, let me generalize this case to a broader class of structures that more closely resembles emergent algebras.

Suppose that $G$ is a group. Then a $G$-multi-antiquandle is an algebraic structure
$(X,(*_{g})_{g\in G})$ that satisfies the following identities:

$x*_{e}y=y$

$x*_{f}(x*_{g}y)=x*_{fg}y$

$(x*_{f}y)*_{g}z=(x*_{g}z)*_{f}(y*_{g}z)$

$x*_{f}x=x$.

whenever $x,y,z\in X,f,g\in G$.

Our artificial intelligence techniques easily generalize to generating $G$-multi-antiquandles even when $G$ is a non-abelian group. The way to do this is to first choose a finitely generated free group $F$ and use our artificial intelligence algorithm to produce an $F$-multi-antiquandle $(X,(*_{f})_{f\in F})$ where $X$ is finite. Then there is a normal subgroup $N$ such that $F/N$ is finite and where we can define an $F/N$-multi-anti-quandle $(X,(*_{fN})_{f\in F/N})$ simply by setting $*_{fN}=*_{f}$ for each $f\in F$.

Suppose that $X$ and $I$ are finite sets. Suppose that $g_{x,i}:X\rightarrow X$ is a permutation for $x\in X,i\in I$ and that $x*_{i}y=g_{x,i}(y)$ whenever $x,y\in X,i\in I$. Then define a loss function $L$ by letting
$$L((X,*_{i}))=|\{(i,j,x,y,z)\mid i,j\in I,x,y,z\in X,(x*_{i}y)*_{j}z\neq(x*_{j}z)*_{i}(y*_{j}z)\}|$$
$$+|\{(i,x)\mid i\in I,x\in X,x*_{i}x\neq x\}.$$

By minimizing the loss function $L$, one should obtain an algebra $(X,(*_{i})_{i\in I})$ with $(X,(*_{i})_{i\in I})=0$. Now, let $F$ be the free group generated by $(z_{i})_{i\in I}$. Then if $v=z_{i_{1}}^{\alpha_{1}}\dots z_{i_{n}}^{\alpha_{n}}$, then define
$x*_{v}y=g_{x,i_{1}}^{\alpha_{1}}\cdots g_{x,i_{n}}^{\alpha_{n}}(y)$. Then
$(X,(*_{v})_{v\in F})$ should hopefully be an $F$-multi-anti-quandle.

I am sure there is an interesting theory behind the $G$-multi-anti-quandles, but I do not think anyone has studied these kinds of structures before.