Concatenations of Linear Orderings and Their Rigidity Characteristics

Source abstract

Among algebraic operations the operation of concatenation for families of words and for orderings is broadly used in various fields of theoretical and applied mathematics, and informatics. It allows to structure to group and to modify information relative required sequences. Degrees of rigidity are considered as a measures of the mobility of a structure with respect to both the automorphism group and definable closures. Classifying structures with respect to rigidity characteristics allows us to determine the dynamic capabilities of the structure when certain elements are fixed. By examining and describing the degrees of rigidity of concatenations of linear orders, we can both classify these concatenations and establish their diversity. We introduce a series of notions relative degrees of rigidity varying their possibilities from absolute rigidity till nowhere almost rigidity, somewhere non-almost rigidity, and absolute finite non-rigidity. These variations of rigidity characteristics are both described and classified for concatenations of linear orderings. We represent a classification of absolutely rigid linear orderings, describe all linear orderings whose all rigidity characteristics equal 1, and describe possibilities of rigidity characteristics for arbitrary concatenations of countable linear orderings.