Concerning similarity transformations of linearly ordered sets
Ben Dushnik, Edwin W. Miller · Bulletin of the American Mathematical Society · 1940
Introduction.As is well known, two linearly ordered sets A and B are said to be similar if there exists a 1-1 correspondence between their elements which preserves order.A function ƒ which defines such a 1-1 correspondence may be called a similarity transformation on A to B. In this note we consider two problems concerning similarity transformations which do not seem to have received attention heretofore.The first problem is the following:(A) Is it true that every infinite ordered set is similar to a proper subset of itself? 2efore stating the second problem we recall a classical theorem concerning well-ordered sets. 3THEOREM.If the set A is well-ordered, and if ƒ is any similarity transformation on A to a subset of A, then f (a) ^ a for every a in A.