Minimality conditions, topologies, and ranks for spherically ordered theories

The class of ordered structures is productively studied both in order to classify them and in various applications connected with comparing of objects and information structuring. Important particular kinds of ordered structures are represented by $o$-minimal, weakly $o$-minimal and circularly minimal ones as well as their variations including definable minimality. We show that the well developed powerful theory for $o$-minimality, circular minimality, and definable minimality is naturally spread for the spherical case. Reductions of spherical orders to linear ones, called the linearizations, and back reconstructions, called the spherifications, are examined. Neighbourhoods for spherically ordered structures and their topologies are studied. It is proved that related topological spaces can be $T_0$-spaces, $T_1$-spaces and Hausdorff ones. These cases are characterized by the cardinality estimates of the universe. Definably minimal linear orders, their definably minimal extensions and restrictions as well as spherical ones are described. The notion of convexity rank is generalized for spherically ordered theories, and values for the convexity rank are realized in weakly spherically minimal theories which are countably categorical.