On Properties Preserved via Types and Tuples of Relations

In mathematics, many natural properties influence or are obtained as a result of the influence of various types of preservation of certain attributes. This applies to geometric, algebraic, logical, dynamic, probabilistic and other mathemat-ical objects that allow us to reveal and classify relationships of the surrounding reality, define new structures and build new significant systems that make it possible to solve theoretical and applied problems. Among these constructions, the fundamental ones are substructures, quotients, filtered products of structures, their combinations, combinations with respect to given families of predicates and equivalence relations, semantic and syntactic generic constructions, etc. Various kinds of preservation of properties were used in the classification of countable models of complete theories and algebras of binary formulae. We study general possibilities of preservation of properties via types producing natural classes of relations. We apply a general approach and characterize properties of reflexivity, non-reflexivity, irreflexivity, symmetry, asymmetry, antisymmetry, transitivity, non-transitivity, and linearity in terms of preservation via appropriate types, together with the properties of orders, preorders and equivalence relations. In terms of preservation, we prove characterizations for precomplete, complete, pre-dense, and dense binary relations, together with linear and dense orders. Besides, we characterize the existence of contours and the projectivity property. Natural generalizations of these properties, axiomatizing spherical orders, are also characterized, forming a description of spherical orders by their preservations. A series of open problems that naturally arise under the consideration of various kinds of type-preservation is posed.