Star-operations on semigroups |
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Authors: | Myeong Og Kim Dong Je Kwak Young Soo Park |
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Institution: | (1) Department of Mathematics Kyungpook National University Taegu, 702-701, Korea yngspark@knu.ac.kr, KR |
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Abstract: | Let S be a grading monoid with quotient group q(S) , let F(S) be the set of fractional ideals of S . For A ∈ F(S) , define A
w
= {x ∈ q(S) \mid J+x \subseteq A for some f.g. ideal J of S with J
-1
=S} and A_ \overline w ={x ∈ q(S)\mid J+x \subseteq A for some ideal J of S with J
-1
=S} . Then w and \overline w are star-operations on F(S) such that w ≤ \overline w . Using these star-operations, we give characterizations of Krull semigroups and pre-Krull semigroups. Also we show that
for every maximal * -ideal P of S , if S
P
is a valuation semigroup, then * -cancellation ideals are * -locally principal ideals, where * is a star-operation on S of finite character. Finally, we show that S is a pre-Krull semigroup (H-semigroup) if and only if the polynomial semigroup Sx] is a pre-Krull semigroup (H-semigroup).
October 15, 1999 |
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Keywords: | |
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