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 theoperation $\ast $ of a finite quasigroup $Q$ is a Latin square on theelements 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 $ afunction $e_{l}$ on $Q^{+}$ by $\alpha =a_{1}a_{2}\cdots a_{n}\mapstoe_{l}(\alpha )=b_{1}b_{2}\cdots b_{n}$ $\Leftrightarrowb_{i+1}=b_{i}\ast a_{i+1}$ 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 paperunder review call $e_{l}$ an $e$-transformation with leader $l$. Thecomposition $E_{k}=e_{l_{1}}\circ e_{l_{2}}\circ \dots \circe_{l_{k}}$ is called an $E$-transformation with leaders $l_{i}$.Indeed the string $E_{k}(\alpha )$ would have a pattern different fromthat of $\alpha $. The authors show that as $\alpha \mapstoE_{k}(\alpha )$ the period changes at least by $k$. It turns out, as aresult of experiments with $E$-transformations of random walks on anintegral torus of a finite size, that for some quasigroups the periodchange could be exponential. The authors make this a basis for theirempirical classification of quasigroups as linear orexponential. (They use a software package which is available at\url{http://twins.ii.edu.mk/trw}. The reader may need to know how to useit. I was able to generate quasigroups. For larger order, the columnsseem to get mixed up.) According to the authors, quasigroups of primeorder have a better chance of being exponential, though some arelinear. The authors mention that processing of strings overexponential quasigroups could be used in producing quasi-pseudo-randomsequences and the linear quasigroups can be used in designtheory. There is, however, the thought that, as the authors mention onpage 68, the choice of parameters and regions (on the torus) caninfluence the classification.