부산대학교 도서관

A quasigroup is a nonempty set $Q$ closed under a binary operation $\ast $ such that for each pair $u,v\in Q$ there exists a unique pair $x,y\in Q$ such that $u\ast x=y\ast u=v$. The table $T$ defining the operation $\ast $ of a finite quasigroup $Q$ is a Latin square on the elements of $Q$. Let $\alpha =a_{1}a_{2}\cdots a_{n}$ be a word (string) with $a_{i}\in Q$. A string $\alpha =a_{1}a_{2}\cdots a_{n}$ is said to have a period $p$ if $a_{i+1}a_{i+2}\cdots a_{i+p}=$ $a_{i+p+1}a_{i+p+2}\cdots a_{i+2p}$ for $i\geq 0$. Denote by $Q^{+}$ the set of nonempty strings on $Q$. Define for some $l\in Q $ a function $e_{l}$ on $Q^{+}$ by $\alpha =a_{1}a_{2}\cdots a_{n}\mapsto e_{l}(\alpha )=b_{1}b_{2}\cdots b_{n}$ $\Leftrightarrow b_{i+1}=b_{i}\ast a_{i+1}$ where $i=0,1,2,\dots ,n-1$ and $l=b_{0}$, and $b_{i}\ast a_{i+1}$ is computed from $T.$ The authors of the paper under review call $e_{l}$ an $e$-transformation with leader $l$. The composition $E_{k}=e_{l_{1}}\circ e_{l_{2}}\circ \dots \circ e_{l_{k}}$ is called an $E$-transformation with leaders $l_{i}$. Indeed the string $E_{k}(\alpha )$ would have a pattern different from that of $\alpha $. The authors show that as $\alpha \mapsto E_{k}(\alpha )$ the period changes at least by $k$. It turns out, as a result of experiments with $E$-transformations of random walks on an integral torus of a finite size, that for some quasigroups the period change could be exponential. The authors make this a basis for their empirical classification of quasigroups as linear or exponential. (They use a software package which is available at \url{http://twins.ii.edu.mk/trw}. The reader may need to know how to use it. I was able to generate quasigroups. For larger order, the columns seem to get mixed up.) According to the authors, quasigroups of prime order have a better chance of being exponential, though some are linear. The authors mention that processing of strings over exponential quasigroups could be used in producing quasi-pseudo-random sequences and the linear quasigroups can be used in design theory. There is, however, the thought that, as the authors mention on page 68, the choice of parameters and regions (on the torus) can influence the classification.