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Factoids #1 (#222)

Jonathan Frech

IV) Commutative, non-associative operations

For any natural num­ber 𝑛, let $\mathrm{Op}_n:=\left\{\star:\mathbb{Z}^2_n\to\mathbb{Z}_n\right\}$$\mathrm{Op}_n:=\left\{\star:\mathbb{Z}^2_n\to\mathbb{Z}_n\right\}$ denote the set of all operations on a set of that order. An operation ★ shall be called com­mu­ta­tive iff $\mathrm{commut}(\star):\Leftrightarrow\forall\,x,y\in\mathbb{Z}_n:x\star y=y\star x$$\mathrm{commut}(\star):\Leftrightarrow\forall\,x,y\in\mathbb{Z}_n:x\star y=y\star x$ and be called associative iff $\mathrm{assoc}(\star):\Leftrightarrow\forall\,x,y,z\in\mathbb{Z}_n:x\star(y\star z)=(x\star y)\star z$$\mathrm{assoc}(\star):\Leftrightarrow\forall\,x,y,z\in\mathbb{Z}_n:x\star(y\star z)=(x\star y)\star z$ holds.

With the above defined, one may study $\mathrm{CnA}_n:=\{\star\in\mathrm{Op}_n:\mathrm{commut}(\star)\land\lnot\mathrm{assoc}(\star)\}$$\mathrm{CnA}_n:=\{\star\in\mathrm{Op}_n:\mathrm{commut}(\star)\land\lnot\mathrm{assoc}(\star)\}$. For 𝑛 = 2, this set is nonempty for the first time, containing a manageable two elements, by name

$$\mathrm{CnA}_2=\Big\{\mathrm{nor}:(x,y)\mapsto 1+xy,\quad\mathrm{nand}:(x,y)\mapsto (1+x)\cdot(1+y)\Big\}.$$$$\mathrm{CnA}_2=\Big\{\mathrm{nor}:(x,y)\mapsto 1+xy,\quad\mathrm{nand}:(x,y)\mapsto (1+x)\cdot(1+y)\Big\}.$$

However, based on the superexponential nature of $\#\mathrm{Op}_n=\#\mathbb{Z}_n^{\mathbb{Z}_n^2}=\#\mathbb{Z}_n^{{\#\mathbb{Z}_n}^2}=n^{n^2}$$\#\mathrm{Op}_n=\#\mathbb{Z}_n^{\mathbb{Z}_n^2}=\#\mathbb{Z}_n^{{\#\mathbb{Z}_n}^2}=n^{n^2}$, the sequence $\mathrm{A079195}_n:=\#\mathrm{CnA}_n$$\mathrm{A079195}_n:=\#\mathrm{CnA}_n$ likely also grows rather quickly, OEIS on­ly listing four members;

$$\mathrm{A079195}=(0,2,666,1\,047\,436,\dots).$$$$\mathrm{A079195}=(0,2,666,1\,047\,436,\dots).$$

Based on this limited numerical evidence, I would suspect the com­mu­ta­tive yet non-associative operations to be rather sparse, i. e.

$$\lim\limits_{n\to\infty}\mathrm{A079195}_n\cdot\left(\#\mathrm{Op_n}\right)^{-1}=0\mod\square.$$$$\lim\limits_{n\to\infty}\mathrm{A079195}_n\cdot\left(\#\mathrm{Op_n}\right)^{-1}=0\mod\square.$$

Analysis source: factoids-1_operations.hs

(Non-)com­mu­ta­tive and (non-)associative operations have also been studied nearly twenty years ago by Christian van den Bosch, author of OEIS sequence A079195. Unfortunately, their site appears to be down, which is where they hosted closed binary operations on small sets (resource found on web.archive.org).

V) Arbitrary polynomial extremum difference

Let 𝜀 > 0 be an arbitrary distance, define $g:=-4x^4-x^3+8x^2+3x-4$$g:=-4x^4-x^3+8x^2+3x-4$. Then $f:=\sfrac{\epsilon}{4}\cdot g$$f:=\sfrac{\epsilon}{4}\cdot g$ has two local maxima at - 1 and 1, whose vertical distance is 𝜀.

VI) Digit sum roots

It holds that $\mathrm{ds}_{10}(108^{12})=108$$\mathrm{ds}_{10}(108^{12})=108$.