Almost quasi-Urbanik structures and theories

The notions of almost quasi-Urbanik structures and theories, and studied possibilities for the degrees of quasi-Urbanikness, both for existential and universal cases were introduced. Links of these characteristics and their possible values are described. These values for structures of unary predicates, equivalence relations, linearly ordered, preordered and spherically ordered structures and theories as well as for strongly minimal ones, and for some natural operations including disjoint unions and compositions of structures and theories were studied. A series of examples illustrates possibilities of these characteristics.