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A characterization of seminormal C-monoids.

Alfred Geroldinger¹, Qinghai Zhong¹

¹Institute for Mathematics and Scientific Computing, NAWI Graz, University of Graz, Heinrichstraße 36, 8010 Graz, Austria.

Bollettino Della Unione Matematica Italiana (2008)

|March 28, 2020

Summary

A C-monoid is seminormal if its reduced class semigroup is a union of groups. This finding establishes a criterion for when seminormal C-monoids are half-factorial, linking semigroup structure to factorization properties.

Keywords:

C-monoids Class semigroups Half-factorial Krull monoids Seminormal

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Area of Science:

Algebraic Theory
Semigroup Theory
Number Theory

Background:

C-monoids are characterized by their reduced class semigroup.
Complete integral closure relates to the semigroup being a group, implying Krull monoid properties.
Seminormality in C-monoids requires further investigation regarding its semigroup structure.

Purpose of the Study:

To characterize seminormal C-monoids using their reduced class semigroup.
To establish a criterion for half-factorial behavior in seminormal C-monoids based on semigroup properties.

Main Methods:

Investigating the relationship between C-monoid properties and their reduced class semigroups.
Utilizing characterizations of C-monoids, Krull monoids, and half-factorial monoids.
Developing criteria based on the structure of the class semigroup.

Main Results:

A C-monoid is seminormal if and only if its reduced class semigroup is a union of groups.
A criterion is established to determine when seminormal C-monoids are half-factorial, based on the class semigroup structure.

Conclusions:

The structure of the reduced class semigroup provides a key insight into the seminormality of C-monoids.
The study connects semigroup theory with factorization properties in algebraic number theory, specifically for C-monoids.