Boundary-Border Extensions of the Kuratowski Monoid

Abstract

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.