On Delta Sets of Some Pseudo-Symmetric Numerical Semigroups with Embedding Dimension Three

Abstract

Let 𝑆� be a numerical semigroup. The catenary degree of an element 𝑠� in 𝑆� is a nonnegative integer used to measure the distance between factorizations of 𝑠�. The catenary degree of the numerical semigroup 𝑆� is obtained at the maximum catenary degree of its elements. The maximum catenary degree of 𝑆� is attained via Betti elements of 𝑆� with complex properties. The Betti elements of 𝑆� can be obtained from all minimal presentations of 𝑆�. A presentation for 𝑆� is a system of generators of the kernel congruence of the special factorization homomorphism. A presentation is minimal if it can not be converted to another presentation, that is, any of its proper subsets is no longer a presentation. The Delta set of 𝑆� is a factorization invariant measuring the complexity of sets of the factorization lengths for the elements in 𝑆�. In this study, we will mainly express the given above invariants of a special pseudosymmetric numerical semigroup family in terms of its generators.