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University of Minnesota

1. Csar, Sebastian Alexander. Root and weight semigroup rings for signed posets.

Degree: PhD, Mathematics, 2014, University of Minnesota

We consider a pair of semigroups associated to a signed poset, called the root semigroup and the weight semigroup, and their semigroup rings, \Rprt and \Rpwt, respectively.Theorem 4.1.5 gives generators for the toric ideal of affine semigroup rings associated to signed posets and, more generally, oriented signed graphs. These are the subrings of Laurent polynomials generated by monomials of the form ti± 1,ti± 2,ti± 1tj± 1. This result appears to be new and generalizes work of~\citet*{BoussicaultFerayLascouxReiner2012},~\citet*{GitlerReyesVillarreal2010} and~\citet{Villarreal1995}. Theorem 4.2.12 shows that strongly planar signed posets P have rings \Rprt, \Rrt{\Pc} which are complete intersections, with Corollary 4.2.20 showing how to compute ΨP in this case. Theorem 5.2.3 gives a Gröbner basis for the toric ideal of \Rpwt in type B, generalizing~\citet*[Proposition 6.4]{FerayReiner2012}. Theorems 5.3.10 and 5.3.21 giving two characterizations (via forbidden subposets versus via inductive constructions) of the situation where this Gröbner basis gives a complete intersection presentation for its initial ideal, generalizing~\citet*[Theorems 10.5, 10.6]{FerayReiner2012

Subjects/Keywords: Semigroups; Signed posets; Toric ideals; Mathematics

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APA (6th Edition):

Csar, S. A. (2014). Root and weight semigroup rings for signed posets. (Doctoral Dissertation). University of Minnesota. Retrieved from http://hdl.handle.net/11299/167039

Chicago Manual of Style (16th Edition):

Csar, Sebastian Alexander. “Root and weight semigroup rings for signed posets.” 2014. Doctoral Dissertation, University of Minnesota. Accessed October 25, 2020. http://hdl.handle.net/11299/167039.

MLA Handbook (7th Edition):

Csar, Sebastian Alexander. “Root and weight semigroup rings for signed posets.” 2014. Web. 25 Oct 2020.

Vancouver:

Csar SA. Root and weight semigroup rings for signed posets. [Internet] [Doctoral dissertation]. University of Minnesota; 2014. [cited 2020 Oct 25]. Available from: http://hdl.handle.net/11299/167039.

Council of Science Editors:

Csar SA. Root and weight semigroup rings for signed posets. [Doctoral Dissertation]. University of Minnesota; 2014. Available from: http://hdl.handle.net/11299/167039

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