Boundary-Border Extensions of the Kuratowski Monoid

The Kuratowski monoid $\mathbf{K}$ is generated under operator composition by closure and complement in a nonempty topological space. It satisfies $2\le \mathrm{\mid }\mathbf{K}\mathrm{\mid }\le 14$. The Gaida-Eremenko (or GE) monoid $\mathbf{K}\mathbf{F}$ extends $\mathbf{K}$ by adding the boundary operator. It satisfies $4\le \mathrm{\mid }\mathbf{K}\mathbf{F}\mathrm{\mid }\le 34$. We show that when $\mathrm{\mid }\mathbf{K}\mathrm{\mid }<14$ the GE monoid is determined by $\mathbf{K}$. When $\mathrm{\mid }\mathbf{K}\mathrm{\mid }=14$ if the interior of the boundary of every subset is clopen, then $\mathrm{\mid }\mathbf{K}\mathbf{F}\mathrm{\mid }=28$. This defines a new type of topological space we call $Kuratowskidisconnected$. Otherwise $\mathrm{\mid }\mathbf{K}\mathbf{F}\mathrm{\mid }=34$. When applied to an arbitrary subset the GE monoid collapses in one of $70$ possible ways. We investigate how these collapses and $\mathbf{K}\mathbf{F}$ interdepend, settling two questions raised by Gardner and Jackson. Computer experimentation played a key role in our research.