Let S and T be numerical semigroups and let k be a positive integer. We say that S is the quotient of T by k if an integer x belongs to S if and only if kx belongs to T. Given any integer k larger than 1 (resp., larger than 2), every numerical semigroup S is the quotient T/k of infinitely many symmetric (resp., pseudo-symmetric) numerical semigroups T by k. Related examples, probabilistic results, and applications to ring theory are shown.
Given an arbitrary positive integer k, it is not true in general that every numerical semigroup S is the quotient of infinitely many numerical semigroups of maximal embedding dimension by k. In fact, a numerical semigroup S is the quotient of infinitely many numerical semigroups of maximal embedding dimension by each positive integer k larger than 1 if and only if S is itself of maximal embedding dimension. Nevertheless, for each numerical semigroup S, for all sufficiently large positive integers k, S is the quotient of a numerical semigroup of maximal embedding dimension by k. Related results and examples are also given.
| Identifer | oai:union.ndltd.org:UTENN/oai:trace.tennessee.edu:utk_graddiss-1564 |
| Date | 01 May 2010 |
| Creators | Smith, Harold Justin |
| Publisher | Trace: Tennessee Research and Creative Exchange |
| Source Sets | University of Tennessee Libraries |
| Detected Language | English |
| Type | text |
| Format | application/pdf |
| Source | Doctoral Dissertations |
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