论文标题
kuratowski monoid的边界 - 边界扩展
Boundary-Border Extensions of the Kuratowski Monoid
论文作者
论文摘要
kuratowski monoid $ \ mathbf {k} $是在操作员组成下通过闭合而生成的，并在非空置拓扑空间中进行补充。它满足$ 2 \ leq | \ mathbf {k} | \ leq14 $。通过添加边界运算符，Gaida-eremenko（或GE）MONOID $ \ MATHBF {kf} $扩展了$ \ Mathbf {K} $。它满足$ 4 \ leq | \ mathbf {kf} | \ leq34 $。我们表明，当$ | \ mathbf {k} | <14 $ ge monoid由$ \ mathbf {k} $确定。当$ | \ Mathbf {k} | = 14 $如果每个子集的边界的内部都粘在一起，则$ | \ Mathbf {kf} | = 28 $。这定义了一种新型的拓扑空间，我们称为$ kuratowski \断开连接$。否则$ | \ Mathbf {kf} | = 34 $。当应用于任意子集时，GE MONOID以70美元的可能方式之一倒塌。我们研究了这些崩溃和$ \ mathbf {kf} $相互依存的如何解决，这解决了Gardner和Jackson提出的两个问题。计算机实验在我们的研究中起着关键作用。
The Kuratowski monoid $\mathbf{K}$ is generated under operator composition by closure and complement in a nonempty topological space. It satisfies $2\leq|\mathbf{K}|\leq14$. The Gaida-Eremenko (or GE) monoid $\mathbf{KF}$ extends $\mathbf{K}$ by adding the boundary operator. It satisfies $4\leq|\mathbf{KF}|\leq34$. We show that when $|\mathbf{K}|<14$ the GE monoid is determined by $\mathbf{K}$. When $|\mathbf{K}|=14$ if the interior of the boundary of every subset is clopen, then $|\mathbf{KF}|=28$. This defines a new type of topological space we call $Kuratowski\ disconnected$. Otherwise $|\mathbf{KF}|=34$. When applied to an arbitrary subset the GE monoid collapses in one of $70$ possible ways. We investigate how these collapses and $\mathbf{KF}$ interdepend, settling two questions raised by Gardner and Jackson. Computer experimentation played a key role in our research.
