Equicardinality on linear orders

Linear orders are of inherent interest infinite model theory, especially in descriptive complexity theory. Here, the class of ordered structures is approached from a novel point of view, using generalized quantifiers as a means of analysis. The main technical result is a characterization of the cardinality quantifiers which can express equicardinality on ordered structures. This result can be viewed as a dichotomy: the cardinality quantifier either shows a lot of periodicity, or is quite non-periodic, the equicardinality quantifier being definable only in the latter case. The main result shows, once more, that there is a drastic difference between definability among ordered structures and definability on unordered structures. Connections of the result to the descriptive complexity of low-level complexity classes are discussed.