Homogenization of a nonsymmetric embedding-dimension-three numerical semigroup

Abdelnaby Taha, Seham; García-Sánchez, Pedro

doi:10.2140/involve.2014.7.77

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Homogenization of a nonsymmetric embedding-dimension-three numerical semigroup

Seham Abdelnaby Taha and Pedro A. García-Sánchez

Vol. 7 (2014), No. 1, 77–96

DOI: 10.2140/involve.2014.7.77

Abstract

Let $n_{1}, n_{2}, n_{3}$ be positive integers with $gcd (n_{1}, n_{2}, n_{3}) = 1$ . For $S = 〈 n_{1}, n_{2}, n_{3} 〉$ nonsymmetric, we give an alternative description, using elementary techniques, of a minimal presentation of its homogenization $\bar{S} = 〈 (1, 0), (1, n_{1}), (1, n_{2}), (1, n_{3}) 〉$ . As a consequence, we show that this minimal presentation is unique. We recover Bresinsky’s characterization of the Cohen–Macaulay property of $\bar{S}$ and present a procedure to compute all possible catenary degrees of the elements of $\bar{S}$ .

Keywords

numerical semigroup, catenary degree, projective monomial curve, homogeneous catenary degree

Mathematical Subject Classification 2010

Primary: 20M14, 20M25

Milestones

Received: 25 February 2013

Revised: 2 May 2013

Accepted: 1 June 2013

Published: 24 October 2013

Communicated by Scott T. Chapman

Authors

Seham Abdelnaby Taha
Departamento de Álgebra Facultad de Ciencias Universidad de Granada Av. Fuentenueva, s/n 18071 Granada Spain

Pedro A. García-Sánchez
Departamento de Álgebra Facultad de Ciencias Universidad de Granada Av. Fuentenueva, s/n 18071 Granada Spain

Vol. 7, No. 1, 2014