Similar sets

A generalization of the elementary geometrical concept of a similarity. Two sets $A$ and $B$ that are totally ordered by relations $R$ and $S$ respectively are said to be similar if there exists a bijection $f : A \rightarrow B$ such that for any $x,y \in A$ it follows from $x\, R\, y$ that $f(x)\, S\, f(y)$.

Comments

An equivalence class of totally ordered sets under the relation of similarity is often called an order type.