Abstract. We study monomial curves, toric ideals and monomial algebras associated to 4-generated pseudo symmetric numerical semigroups. Namely, we determine indispensable binomials of these toric ideals, give a characterization for these monomial algebras to have strongly indispensable minimal graded free resolutions. We also characterize when the tangent cones of these monomial curves at the origin are Cohen-Macaulay.

In this article, we give a fast and an easily implementable algorithm for computing the Arf closure of an irreducible algebroid curve (or a branch). Moreover, we study the relation between the branches having the same Arf closure and their regularity indices. We give some results and a conjecture, which are steps towards the interpretation of Arf closure as a specific way of taming the singularity.

We compute Betti numbers for a Cohen–Macaulay tangent cone of a monomial curve in the affine $4$-space corresponding to a pseudo-symmetric numerical semigroup. As a byproduct, we also show that for these semigroups, being of homogeneous type and homogeneous are equivalent properties.

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