Variations of rigidity

One of the main derived objects of a given structure is its automorphism group, which shows how freely elements of the structure can be related to each other by automorphisms. Two extremes are observed here: the automorphism group can be transitive and allow any two elements to be connected to each other, or can be one-element, when no two different elements are connected by automorphisms, i.e., the structure is rigid. The rigidity given by a one-element group of automorphisms is called semantic. It is of interest to study and describe structures that do not differ much from semantically rigid structures, i.e., become semantically rigid after selecting some finite set of elements in the form of constants. Another, syntactic form of rigidity is based on the possibility of getting all elements of the structure into a definable closure of the empty set. It is also of interest here to describe “almost” syntactically rigid structures, i.e., structures covered by the definable closure of some finite set. The paper explores the possibilities of semantic and syntactic rigidity. The concepts of the degrees of semantic and syntactic rigidity are defined, both with respect to existence and with respect to the universality of finite sets of elements of a given cardinality. The notion of a rigidity index is defined, which shows an upper bound for the cardinalities of algebraic types, and its possible values are described. Rigidity variations and their degrees are studied both in the general case, for special languages, including the one-place predicate signature, and for some natural operations with structures, including disjunctive unions and compositions of structures. The possible values of the degrees for a number of natural examples are shown, as well as the dynamics of the degrees when taking the considered operations.