The Galaxies of Nonstandard Enlargements of Infinite and Transfinite grpahs of Higher Ranks: II

Abstract

This report is an improvement of a prior report (Report 814). It sharpens the principal theorem (Theorem 5.1 of Report 814) and also simplifies its proof. There are also several minor changes involving clarifications and corrections of misprints. The prior abstract remins the same as follows: In a previous work, the galaxies of the nonstandard enlargements of connected, conventionally infinite and also of transfinite graphs of the first rank of transfiniteness were defined, examined, and illustrated by some examples. In this work it is shown how the results of the prior work extend to transfinite graphs of higher ranks. Among those results are following: Any such enlargement either has exactly one galaxy, its principal one, ot it has infinitely many such galaxies. In the latter case, the galaxies are partially ordered by their "closeness" to the principal galaxy. Also, certain sequences of galaxies whose members are totally ordered by that "closeness" property are identified.