Let $X$ be a finite set. Recall that a binary operation $\ast$ on $X$ is said to be a quasigroup operation if there are binary operations $/,\backslash$ where $(x/y)\ast y=(x\ast y)/y=x$ and $x\ast(x\backslash y)=x\backslash(x\ast y)=y$.
Let $T:X^2\rightarrow X^2$ be a bijection. Let $\ast_{l,+},\ast_{l,-},\ast_{r,+},\ast_{r,-}$ be the binary operations where $T(x,y)=(x\ast_{l,+}y,x\ast_{r,+}y),T^{-1}(x,y)=(x\ast_{l,-}y,x\ast_{r,-}y)$ whenever $x,y\in X$. Then we say that $T$ is a well-mixing if each of the operations $\ast_{l,+},\ast_{l,-},\ast_{r,+},\ast_{r,-}$ is a quasigroup operation.
We say that two well-mixing algebras $S:X^2\rightarrow X^2,T:X^2\rightarrow X^2$ are isotopic if there exists permutations $f_1,\dots,f_4:X\rightarrow X$ along with a $\sigma,\tau:X^2\rightarrow X^2$ where $S=\tau\circ(f_1\times f_2)\circ T\circ(f_3\times f_4)\circ\sigma$ and where $\sigma,\tau\in\{1_{X^2},\text{SW}_X\}$ where $\text{SW}_X(x,y)=(y,x)$ for all $x,y$.
Let $t_n$ denote the number of well-mixing algebras $(X,T)$ up to isotopy where $|X|=n$. I would like to know the exact values of $t_n$ for small $n$. The number of quasigroups up to isotopy has been calculated for $n\leq 11$.
I am interested in a classification of small well-mixing algebras since one can use these permutations to construct block ciphers and similar cryptographic functions.
