论文标题
五角大楼方程的固定理论解决方案
Set-theoretic solutions of the Pentagon Equation
论文作者
论文摘要
五边形方程在非空置集$ s $上的设置理论解决方案是$ s \ s \ colon s^2 \ to s^2 $，这样$ s_ {23} s_ {13} s_ {13} s_ {12} = s_ {12} s_ {12} s_ {23} $ s_ {23} = \ MATHRM {id} \ times s $和$ s_ {13} =（τ\ times \ mathrm {id}））翻转地图，即$τ（x，y）=（y，x）$。我们给出所有参与解决方案的描述，即$ s^2 = \ mathrm {id} $。结果表明，此类解决方案是由$ s $作为直接产品$ x \ times a \ times g $和地图$σ\ colon a \ to \ mathrm {sym}（x）$确定的，其中$ x $是非空的套装，$ a，$ a，g $是基本的阿贝安$ 2 $ -2 $ -2 $ -2 $ -2 $ -2 $。同构解决方案取决于$ a $ a $，$ g $和$ x $的基础性，即地图$σ$无关紧要。特别是，如果$ s $是基数有限的$ 2^n（2m+1）$，对于某些$ n，m \ geq 0 $，那么，在$ s $上，五角大楼方程的$ s $ cys $ s $ cy of $ n，m \ geq 0 $。
A set-theoretic solution of the Pentagon Equation on a non-empty set $S$ is a map $s\colon S^2\to S^2$ such that $s_{23}s_{13}s_{12}=s_{12}s_{23}$, where $s_{12}=s\times\mathrm{id}$, $s_{23}=\mathrm{id}\times s$ and $s_{13}=(τ\times\mathrm{id})(\mathrm{id}\times s)(τ\times\mathrm{id})$ are mappings from $S^3$ to itself and $τ\colon S^2\to S^2$ is the flip map, i.e., $τ(x,y) =(y,x)$. We give a description of all involutive solutions, i.e., $s^2=\mathrm{id}$. It is shown that such solutions are determined by a factorization of $S$ as direct product $X\times A \times G$ and a map $σ\colon A\to\mathrm{Sym}(X)$, where $X$ is a non-empty set and $A,G$ are elementary abelian $2$-groups. Isomorphic solutions are determined by the cardinalities of $A$, $G$ and $X$, i.e., the map $σ$ is irrelevant. In particular, if $S$ is finite of cardinality $2^n(2m+1)$ for some $n,m\geq 0$ then, on $S$, there are precisely $\binom{n+2}{2}$ non-isomorphic solutions of the Pentagon Equation.
