Partial order on a family of image-subsets of a linearly ordered set Original Research Article

For image-subsets image of the rationals image, define image if image holds for at least image ordered pairs image, where image are integers, image. We prove that (1) the relation image is transitive if and only if image, and (2) there is a cyclic sequence image of image-subsets of image if and only if image. We also investigate the length of such cyclic sequences.