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 $2\leq|\mathbf{K}|\leq14$. 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.Comment: 48 pages, 9 figure

The Encyclopedia of Neutrosophic Researchers, 5th Volume

Neutrosophic set, neutrosophic logic, neutrosophic probability, neutrosophic statistics, neutrosophic measure, neutrosophic precalculus, neutrosophic calculus and so on are gaining significant attention in solving many real life problems that involve uncertainty, impreciseness, vagueness, incompleteness, inconsistent, and indeterminacy. In the past years the fields of neutrosophics have been extended and applied in various fields, such as: artificial intelligence, data mining, soft computing, decision making in incomplete / indeterminate / inconsistent information systems, image processing, computational modelling, robotics, medical diagnosis, biomedical engineering, investment problems, economic forecasting, social science, humanistic and practical achievements. There are about 7,000 neutrosophic researchers, within 89 countries around the globe, that have produced about 4,000 publications and tenths of PhD and MSc theses, within more than two decades. This is the fifth volume of the Encyclopedia of Neutrosophic Researchers, edited from materials offered by the authors who responded to the editor’s invitation, with an introduction contains a short history of neutrosophics, together with links to the main papers and books